APPENDIX 1 A Review of Some Basic Concepts of Topology and Measure Theory
In this appendix we summarize, mainly without proof, some standard results from topology and measure theory. The aims are to establish terminology and notation, to set out results needed at various stages in the text in some specific form for convenient reference, and to provide some brief perspectives on the development of the theory. For proofs and further details the reader should refer, in particular, to Kingman and Taylor (1966), Chapters 1-6, whose development and terminology we have followed rather closely.
Al.l. Set Theory
A set A of a space !![ is a collection of elements or points of !!l. When x is an element of the set A, we write x E A (x belongs to or is included in A). The set of points of!![ not included in A is the complement of A, written A c. If A, B are two sets of points from !![ their union, written A u B, is the set of points in either A or B or both; their symmetric difference, written A/::::,. B, is the set of points in A or B but not both; and their intersection, written A n B, is the set of points in both. If every element of B is also an element of A we say B is included in A (B ~ A), or ·A contains B (A 2 B). In this case the proper difference of A and B, written either A - BorA \B, is the set of points of A but not B. More generally, we use A - B for A n Be, but A - B = A/::::,. B only when A2B. The operations n and /::::,. on subsets of!![ are commutative, associative, and distributive. The class of all such subsets thus forms an algebra with respect to these operations, where 0, the empty set, plays the role of identity for 6, Al.2. Topologies 593
and !!l the role of identity for n. The special relation AnA=A implies that the algebra is Boolean. More generally, any class of sets closed under the operations of n and !::,. is called a ring, or an algebra if !!l itself is a member of the class. A semiring is a class of sets .91 with the properties (i) .91 is closed under intersections, and (ii) every symmetric difference of sets in .91 can be represented as a finite union of disjoint sets in .91. The ring generated by an arbitrary family of sets 1F is the smallest ring containing 17, or, equiv• alently, the intersection of all rings containing !F. Every element in the ring generated by a semiring .91 can be represented as a union of disjoint sets of .91. If qt is a finite ring, there exists a basis of disjoint elements of 9t such that every element in 9t can be represented uniquely as a union of disjoint elements of the basis. The notions of union and intersection can be extended to arbitrary classes of sets. If {An; n = 1, 2, ... } is a sequence of sets, we write An j A = lim An if An£ An+l• n = 1, 2, ... ' and A= Uf An; similarly, if An 2 An+l• n = 1, 2, ... ' we write An! A = lim An if A = nf An. A monotone class is a class of sets closed under monotone increasing sequences. A ring (algebra) that is closed under countable as well as finite unions is called a a-ring (a-algebra). The a-ring generated by a class of sets rl, written a(rl), is the smallest a-ring containing rl. A a-ring is countably generated if it can be generated by a countable class ofrl. The following result, linking a-rings to monotone classes, is useful in identifying the a-ring generated by certain classes of sets.
Proposition Al.l.I (Monotone Class Theorem). If 9t is a ring, and rl is a monotone class containing qt, then rl contains a(!Jt).
A closely related result uses the concept of a Dynkin system: !?) is a Dynkin system if
(i) !!£ E !?); (ii) !?) is closed under proper differences; and (iii) !?) is closed under monotone increasing limits.
Proposition Al.l.II (Dynkin System Theorem). If [I' is a class of sets closed under finite intersections, and !?) is a Dynkin system containing Y, then !?) contains a(Y).
A1.2. Topologies
A topology 1111 on a space !!l is a class of subsets of !!l that is closed under arbitrary unions and finite intersections and that includes the empty set 0 and the whole space !!l; the members of 1111 are open sets, while their comple- 594 APPENDIX 1. A Review of Some Basic Concepts ments are closed sets. The pair (&r, Cl//) is a topological space. The closure of an arbitrary set A from &r, written A, is the smallest closed set (equivalently, the intersection of all closed sets) containing A. The interior of A, written A 0 , is the largest open set (equivalently, the union of all open sets) contained within
A. The boundary of A, written oA, is the difference A\A0 • The following elementary properties of boundaries are needed in the discussion of weak convergence of measures.
Proposition A1.2.1. (a) o(A u B) ~ oA u oB; (b) o(A n B) ~ oA u oB; (c) oAC = oA.
A neighbourhood of the point x E &r with respect to the topology Cllt (or, more briefly, a 1111-neighbourhood of x) is an open set from Cl/t containing x. Cl/t is a Hausdorff or T2 topology if the open sets separate points, that is, if for x =F y, x and y possess disjoint neighbourhoods. A family of sets ff forms a base for the topology 1111 if every U E 1111 can be represented as a union of sets in ff, and ff ~ 1111. 1111 is second countable if it has a countable base, that is, if there exists a countable family ff with these properties. The topology Cl//1 generated by the base !F;_ has the same open sets as, and hence coincides with, the topology 11112 generated by the base ff,_ if every set in /#'1 can be represented as a union of sets in :#'2 , and vice versa. In this case the topological spaces (&r, Cl/t 1 ) and (&r, Cllt 2 ) are said to be equivalent. Given a topology Cl/t on &r, a notion of convergence of sequences (or more generally nets, but we do not need the latter concept) can be introduced by saying xn--+ x in the topology 1111 if, given any Cl//-neighbourhood of x, 1111x, there exists an integer N (depending on the neighbourhood in general) such that Xn E Ux for n ~ N. Conversely, nearly all the important types of convergence can be described in terms of a suitable topology. In this book, the over• whelming emphasis is on metric topologies, where the open sets are defined in terms of a metric or distance function p( ·) satisfying for arbitrary x, y, z E &r the conditions
(i) p(x, y) = p(y, x); (ii) p(x, y) ~ 0 and p(x, y) = 0 if and only if x = y; (iii) (triangle inequality) p(x, y) + p(y, z) ~ p(x, z). With respect to a given distance function p, the open sphere S(x, e) is the set {y: p(x, y) < e}, being defined for any e > 0. The metric topology generated by p is the smallest topology containing the open spheres; it is necessarily Hausdorff. A set is open in this topology if and only if every point in the set can be enclosed by an open sphere lying wholly within the set. A sequence of points {xn} converges to x in this topology if and only if p(xn, x) --+ 0. A limit point y of a set A is a limit of a sequence of points Xn E A, with Xn =I= y; y need not necessarily be in A. The closure of A in the metric topology is the union of A and its limit points. A space &r with topology 1111 is metrizable if a distance A1.2. Topologies 595
function p can be found such that if/1 is equivalent to the metric topology generated by p. Two metrics on the same space X are equivalent if they each generate the same topology on X. A sequence of points {x": n ;;:,: 1} in a metric space is a Cauchy sequence if p(x", xm)--+ 0 as n, m--+ oo. The space is complete if every Cauchy sequence has a limit, that is, if for every Cauchy sequence {x"} there exists a point x EX such that p(x", x)--+ 0. A set ::0 is dense in X if, for every e > 0, every point in X can be approximated by points in ::0, that is, given x E X, there exists d E ::0 such that p(x, d) < e. The space X is separable if there exists a countable dense set, also called a separability set. If X is a separable metric space, the spheres with rational radii and centres on a countable dense set form a countable basis for the topology. Given two topological spaces (X1 , if/11 ) and (X2 , if/12 ), a mapping f( ·)from (X1 , if/1 1 ) into (X2 , if/12 ) is continuous if the inverse imagef-1 (U) of every open set U in if/12 is an open set in if/1 1 • If both spaces are metric spaces, the mapping is continuous if and only if for every x E X1 and every e > 0, there exists c5 > 0 such that p2 (f(x'),f(x)) < e whenever p 1 (x', x) < c5, where Pu p2 are the metrics in Xu X2 , respectively, a statement we can more loosely express by saying f(x')--+ f(x) whenever x' --+ x. A homeomorphism is a 1: 1 continuous• both-ways mapping between two topological spaces. A famous theorem of Urysohn asserts that any complete separable metric space (c.s.m.s.) can be mapped homeomorphically into a countable product of unit intervals. A space that can be mapped homeomorphically into an open subset of a c.s.m.s. is called a Polish space. The theory we develop in Appendix 2 can be carried through for an arbitrary Polish space with only minor changes, but we do not seek this greater generality. A set K in a topological space (X, lf/l) is compact if every covering of K by a family of open sets contains a finite subcovering (K s; U~ U~, U~ E if/1, implies the existence of oc 1 , ... , aN such that K s; Uf U~J It is relatively compact if its closure is compact. In a separable space every open covering contains a countable subcovering and consequently it is sufficient to check the compact• ness property for sequences of open sets rather than general families. More generally, for a c.s.m.s. the following important characterizations of compact sets are equivalent.
Proposition A1.2.11 (Metric Compactness Theorem). Let X be a c.s.m.s. Then the following properties of a subset K of X are equivalent, and each is equivalent to the compactness of K.
(i) (Heine-Borel property) Every countable open covering of K contains a finite subcovering. (ii) (Bolzano-Weierstrass property) Every infinite sequence of points in K contains a convergent subsequence with its limit inK. (iii) (Total boundedness and closure) K is closed, and for every e > 0, K can be covered by a finite number of spheres of radius e. 596 APPENDIX 1. A Review of Some Basic Concepts
(iv) Every sequence {Fn} of closed subsets of K with nonempty finite inter• sections (nf Fn "# 0: the finite intersection property) has nonempty total intersection (n'i' Fn "# 0).
The space f!l" itself is compact if the compactness criterion applies with f!l" in place of K. It is locally compact if every point of f!l" has a neighbourhood with compact closure. A space with a locally compact second countable topology is always metrizable. In a c.s.m.s. local compactness implies a• compactness: the whole space can be represented as a countable union of compact sets (take the compact closures of the neighbourhoods of any count• able dense set). Any finite-dimensional Euclidean space is a-compact, but the same does not apply to infinite-dimensional spaces such as C[O, 1] or the infinite-dimensional Hilbert space t2 • A useful corollary of Proposition A1.2.11 is that any closed subset F of a compact set in a complete metric space is again compact; for by (ii) any infinite sequence of points ofF has a limit point in K, and by closure the limit point is also in F -hence, F is compact.
Al.3. Finitely and Countably Additive Set Functions
Let d be a class of sets in f!l", and e( ·) a real- or complex-valued function defined on d. eo is finitely additive on d if for finite families {A to ••• ' AN} of disjoint sets from d, with their union also in d, there holds
If a similar result holds for sequences of sets {A;: i = 1, 2, ... } then e is countably additive (equivalently, a-additive) on d. A countably additive set function on d is a measure if it is nonnegative; a signed measure if it is real-valued but not necessarily nonnegative; and a complex measure if it is not necessarily real-valued. A determining class for a particular type of set function is a class of sets with the property that if two set functions of the given type agree on the determining class, then they coincide. In this case we can say that the set function is determined by its values on the determining class in question. The following proposition gives two simple results on determining classes. The first is a consequence of the representation of any element in a ring of sets as a disjoint union of the sets in any generating semiring; the second can be proved using a monotone class argument and the continuity lemma A1.3.11 immediately following.
Proposition Al.3.1. (a) A finitely additive, real- or complex-valued set function defined on a ring dis determined by its values on any semiring generating d. A1.3. Finitely and Countably Additive Set Functions 597
(b) A countably additive real- or complex-valued set function defined on a (1-ring [/' is determined by its values on any ring generating f/'.
Proposition Al.3.11 (Continuity Lemma). Let 1-1( ·) be a finite real- or complex• valued, finitely additive set function defined on a ring d. Then fL is countably additive on d if and only if for every decreasing sequence {An: n = 1, 2, ... } of sets with An! 0,
So far we have assumed the set functions to take finite values on all the sets for which they are defined. It is frequently convenient to allow a non• negative set function to take the value +oo; this leads to few ambiguities and simplifies many statements. We then say that a finitely additive set function e(.) defined on an algebra or (j-algebra dis totally finite if, for all finite unions
of disjoint sets A 1 , •.. , AN in d, there exists M < oo such that
N I le
Theorem Al.3.111 (Extension Theorem). A finitely additive, nonnegative set function defined on a ring 9f can be extended to a measure on (j(!J,f) if and only if it is countably additive on &/.
As an example of the use of this theorem we cite the well-known result that a right-continuous monotonic increasing function F( ·) on IR can be used to define a measure on the Borel sets of IR (the sets in the smallest (j-ring containing the intervals) through the following sequence of steps: (i) define a nonnegative set function on the semiring of half-open intervals (a, b] by setting fLF(a, b] = F(b)- F(a); 598 APPENDIX 1. A Review of Some Basic Concepts
(ii) extend ilF by additivity to all sets in the ring generated by such intervals (this ring consisting, in fact, of all finite disjoint unions of such half-open intervals); (iii) establish countable additivity on this ring by appealing to compactness properties of finite closed intervals; and (iv) use the extension theorem to assert the existence of a measure extending the definition of ilF to the a-ring generated by the half-open intervals, that is, the Borel sets.
The intrusion of the topological notion of compactness into this otherwise measure-theoretic sequence is a reminder that in most applications there is a close link between open and measurable sets. Generalizing the corresponding concept for the real line, the Borel sets in a topological space are the sets in the smallest a-ring (necessarily a a-algebra) f!i!'l' containing the open sets. A Borel measure is any measure defined on the Borel sets. The properties of such measures when X is a c.s.m.s. are explored in Appendix 2. Returning to the general discussion, we note that no simple generalization of the extension theorem is known for signed measures. However, there is an important result, which shows that in some respects the study of signed measures can always be reduced to the study of measures.
Theorem A1.3.IV (Jordan-Hahn Decomposition). Let~ be a signed measure defined on a a-algebra!/. Then~ can be written as the difference e = e+- c of two measures ~+, e-, on f/ and X can be written as the union of two disjoint sets u+, u-, in f/ such that for all E e f/ e+(E) =~(En u+) and
~-(E)= -~(En u-), and hence in particular~+ (U-) = ~- (U+) = 0.
The measures~+ and ~- appearing in this theorem are called the upper and lower variations of~. respectively. The total variation of~ is their sum ~(A) =~+(A) + ~-(A). It is clear from the theorem that n(IP') ~(A) = sup L ie(A;)i, IP'(A) 1 where the supremum is taken over all finite partitions IP of A into disjoint measurable sets. Thus, ~is totally bounded if and only if ~(X)< oo. In this case ~(A) acts as a norm on the space of totally bounded signed measures ~ on!/; it is referred to as the variation norm and sometimes written ~(X)= 11~11. A1.4. Measurable Functions and Integrals 599
A1.4. Measurable Functions and Integrals
A measurable space is a pair (¥, ff), where ¥ is the space and ff a er-ring of sets defined on it. A mapping f from a measurable space (¥1 , ffd into a measurable space (¥2 , 31'2 ) is itself measurable if, for all A E 31'2 , f-1 (A) E 31'1 • Note that the inverse images in ¥ 1 of sets in 31'2 form a er-ring r§1, say, and the requirement for measurability is that r§1 s; 31'1 • By specializing to the case that ¥ 2 is the real line~ with§;_ the er-algebra of Borel sets generated by the intervals, the criterion for measurability simpli• fies as follows.
Proposition Al.4.1. A real-valued function f: (¥, ff)-+ ~is measurable if and only if the set {x: f(x) ~ c} is a set in ff for every real c.
The family of real-valued measurable functions on a measurable space (¥, ff) has many striking properties. It is closed under the operations of addition, subtraction, multiplication, and (with due attention to zeros) divi• sion. Moreover, any monotone limit of measurable functions is measurable. If ¥ is also a topological space, and ff the Borel er-field on ¥, then every continuous function on ¥ is measurable. The next proposition provides an important approximation result for measurable functions. Here a simple function is a finite linear combination of indicator functions of measurable sets, that is, a function of the form N s(x) = L ck!Ak(x), 1
where c1 , ... ,eN are real and A 1 , ... ,AN are measurable sets.
Proposition Al.4.11. A nonnegative function f: (¥, ff)-+ ~+ is measurable if and only if it can be represented as the limit of a monotonic increasing sequence of simple functions.
Now let Jl be a measure on ff. We call the triple(¥, 31', JL) a finite or er-finite measure space according to whether Jl has the corresponding property; in the special case of a probability space, when Jl has total mass unity, the triple is more usually written (Q, cff, &'), where the sets of the er-algebra cff are inter• preted as events, a measurable function on (Q, C) is a random variable, and &' is a probability measure. We turn to the problem of defining an integral (or in the probability case an expectation) with respect to the measure JL.If s = Lf ck!Ak is a nonnegative simple function, set
l s(X)JL(dx) = l S djl = f CkJL(Ak), J~ J~ 1 where we allow +oo as a possible value of the integral. Next, for any non- 600 APPENDIX 1. A Review of Some Basic Concepts negative measurable function J, and any sequence of simple functions {sn} approximating f from below, set
r fdJl. =lim r sndJl., J~ n-+ f+(x) = [J(x)]+ = max(f(x), 0), J_(x) = f+(x)- f(x), and, if Jf+dll and Jf_dJl. are both finite (equivalently, J~lfldJl. is finite), say that f is integrable and then define, for any integrable function J, tfdJl. = f~f+dJl.- f~f-dJl.. The resulting abstract Lebesgue integral is well defined, additive, linear, order preserving, and enjoys strikingly elegant continuity properties. These last are set out in the theorem below, where we say fn ~ f Jl.-almost everywhere (Jl.-a.e., or a.e. Jl.) if the (necessarily measurable) set {x: fn(x) + f(x)} has Jl.• measure zero. [In the probability case we refer to almost sure (a.s.) rather than a.e. convergence.] Theorem Al.4.111 (Lebesgue Convergence Theorems). The following results hold for a sequence of measurable functions {J,.; n = 1, 2, ... } defined on the measure space (q', !F, Jl.): (a) (Fatou's lemma) If J, ~ 0, l lim infj,(x)dJl.(X) ~ lim inf l f,.(x)dJl.(X) JPI n-+CX) n-+-oo Jfl' (b) (Monotone convergence theorem) IfJ,. ~ 0, J,. if (a.e. Jl.) then f is measur• able and lim r J..dJl. = r fdJl. n-+ lim r J..dJl. = r fdJl.. n-+ Iff is an integrable function, the indefinite integral off over any measurable subset A can be defined by A1.4. Measurable Functions and Integrals 601 ~~(A)~ Lf dJl ~ t IAf dJl, where IA is the indicator function of A. It is clear that ~~ is totally finite and finitely additive on !/'. Moreover, it follows from the dominated convergence theorem that if An E !/',An! 0, then IAJ-+ 0 and hence ~1 (An)-+ 0. Thus, ~~ is also countably additive, that is, a signed measure on !/'. This raises the question of which signed measures can be represented as indefinite integrals with respect to a given ll· The essential feature is that the ~-measure of a set should tend to zero with the Jl-measure. More specifically, ~ is absolutely continuous with respect to Jl. wherever Jl.(A) = 0 implies ~(A) = 0; we then have the following theorem. Theorem A1.4.IV (Radon-Nikodym Theorem). Let (g[, .1F, Jl.) be a a-finite measure space and ~ a totally finite measure or signed measure on !F. Then there exists a measurable integrable function f, such that (A1.4.1) ~(A) = Lf(x)dJl.(x) (all A E !F), if and only if ~ is absolutely continuous with respect to Jl.; moreover f is a.e. uniquely determined by (A1.4.1), in the sense that any two functions satisfying (A1.4.1) for all A E .1F must be equal (a.e. Jl.). The function f appearing in (A1.4.1) is usually referred to as a Radon• Nikodym derivative of~ with respect to Jl., written d~/dJl.. There is an obvious extension of Theorem A1.4.IV to the case where ~is a-finite; in this extension (Al.4.1) holds for subsets A of each of the denumer• able family of measurable sets on which ~ is totally finite. Finally, we consider the relation between a fixed a-finite measure Jl. and an arbitrary a-finite signed measure~. ~is said to be singular with respect to Jl. if there is a set E in .1F such that Jl.(E) = 0 and for all A E .1F, ~(A) = ~(En A) [so that also ~(£<) = 0 and Jl.(A) = Jl.(A n £<)]. We then have the following theorem. Theorem A1.4.V (Lebesgue Decomposition Theorem). Let (,q[, .1F, Jl.) be a a-finite measure space and~(·) a finite or a-finite signed measure on !F. Then there exists a unique decomposition of~. ~ = ~s + ~ac• into components that are, respectively, singular and absolutely continuous with respect to Jl.. 602 APPENDIX 1. A Review of Some Basic Concepts A1.5. Product Spaces If!!£, iJjJ are two spaces, the Cartesian product !!£ x iJjJ is the set of ordered pairs {(x, y): x e !!£, y e iJjj}. If!!£ and iJjJ are either topological or measure spaces, there is a natural way of combining the original structures to produce a structure in the product space. Let us consider first the topological case. If U, V are neighbourhoods of the points x e !!£, y e iJjJ with respect to topologies o/1, 1/, define a neighbourhood of the pair (x, y) as the product set U x V. The class of product sets of this kind is closed under finite intersections as a consequence of the relation (U x V) n (A x B) = (U n A) x (V n B). It can therefore be taken as the basis of a topology in !!£ x iJjf, which is called the product topology and denoted o/1 ® 1/ [we follow Bremaud (1981), for example, in using a distinctive product sign as a reminder that the product entity here is generated by the elements of the factors]. Most properties enjoyed by the component (or coordinate) topologies are passed on to the product topology. In particular, if!!£, iJjJ are both c.s.m.s, then !!£ x iJjJ is also a c.s.m.s. with respect to any one of a number of equivalent metrics, of which perhaps the simplest is p((x, y), (u, v)) = max{p~(x, u), p~(y, v)}. More generally, if {!!It, t e ff} is a family of spaces, the Cartesian product f£ =X<~) te:T may be defined as the set of all functions x: ff-+ Ut !!It, such that x(t) e !!It. A cylinder set in this space is a set in which restrictions are placed on a finite subset of the coordinates, say on x(t1), ... , x(tN), the values of the other coordinates being unrestricted in their appropriate spaces. A family of basic open sets in f£ can be defined by choosing open sets {Ui s;: !!It,• i = 1, ... , N} and requiring x(ti) e Ui, i = 1, ... , N. The topology generated by the class of cylinder sets of this form is called the product topology in f£. A remarkable property of this topology is that if the coordinate spaces !!It are individually compact in their respective topologies, then fi is compact in the product topology. On the other hand, if the individual !!£, are metric spaces, there are again many ways in which fi can be made into a metric space [e.g., by using the supremum of the distances Pt(x(t), y(t))], but the topologies they generate are not in general equivalent among themselves, or to the product topology defined earlier. Turning now to the measure context, let (!!£,:IF, JJ.) and (iJjj, f§, v) be two measure spaces. The product u-ring :IF ® f§ is the u-ring generated by the semiring of measurable rectangles A x B, with A e :IF, B e f'§. The product measure Jl. x v is the extension to the u-ring of the countably additive set function defined on such rectangles by A1.5. Product Spaces 603 (J.l x v)(A x B) = J.l(A)v(B) and extended by additivity to the ring of all finite disjoint unions of such rectangles. If J.l, v are both finite, then so is J.l x v; similarly, if J.l, v are a-finite so is J.l x v. The product measurable space is the space (~ x 11JJ, ff ® ~}, and the product measure space is the space(~ x OJ/, ff ® ~. J.l x v). All the defini• tions extend easily to the products of finite families of measure spaces. In the probability context, they form the natural framework for the discussion of independence. In the context of integration theory, the most important results pertain to the evaluation of double integrals, the question we take up next. Let Yf = ff ®~and n = J.l x v. If Cis Yf-measurable, its sections Ax= {y: (x, y) E A} and AY = {x: (x, y) E A} are, respectively, ~-measurable for each fixed x and ff -measurable for each fixed y. (The converse to this result, that a set whose sections are measurable is Yf -measurable, is false, however.) Similarly, if f(x, y) is Yf -measurable, then regarded as a function of y, it is ~-measurable for each fixed x, and regarded as a function of x, it is §'-measurable for each fixed y. Introducing integrals with respect to J.l, v, write r f(x, y)dv(y) if the integrand is v-integrable, s(x) = {Jqlf +oo otherwise; r f(x, y)dJ.l(X) if the integrand is J.l-integrable, t(y) = {J~ +oo otherwise. We then have the following theorem. Theorem A1.5.1 (Fubini's Theorem). Let (~, ff, J.l) and (OJ/, ~. v) be u-finite measure spaces, and let ( !Z, Yf, n) denote the product measure space. (a) Iff is Yf -measurable and n-integrable, then s(x) is ff -measurable and j.l-integrable, t(y) is ~-measurable and v-integrable, and L/ dn = t s dJ.l = fqlf t dv. (b) Iff is Yf-measurable and f;;::: 0, it is necessary and sufficient for f to be n-integrable that either s be J.l-integrable or t be v-integrable. Not all the important measures on a product space are product measures; in the probability context, in particular, it is necessary to study general bivariate probability measures and their relations to the marginal and con• ditional measures they induce. Thus, if n is a probability measure on (~ x 11JJ, ff ® ~), we define the marginal probability measures n~ and 1tqlj to be the projections of n onto (~, ff) and (OJ/, ~), respectively, that is, the 604 APPENDIX 1. A Review of Some Basic Concepts measures defined by nil'(A) = n(A x (A1.5.1) n(A x B) = Ldnil'(x)Q(Bix), where Q(Bix) may be regarded as the conditional probability of observing the event B given the occurrence of x. Such a family is also known as a dis• integration of n. Proposition Al.5.11. Given a family {Q( ·ix), x E .9r} of probability measures on ( (A1.5.2) Indeed, the integral at (Al.5.1) is not defined unless Q(BI·) is ~-measurable. When it is, the right side of (Al.5.2) can be extended to a finitely additive set function on the ring of finite disjoint unions of rectangle sets. Countable additivity and the extension to a measure for which (A1.5.2) holds then follow along standard lines using monotone approximation arguments. The projection of n onto the space ( (a.e. na-), respectively, but because there are nondenumerably many such relations to be checked, it is not obvious that the exceptional sets of measure zero can be combined into a single such set. The problem, in fact, is formally identical to that considered in Chapter 6, Theorem 6.l.VI, and the arguments developed there can be applied equally here. They rest, however, on the assumption of additional topological properties for o/J, and we therefore obtain only the following partial converse to Proposition Al.5.11. Proposition Al.S.III (Existence of Regular Conditional Probabilities). Let (o/J, ~) be a c.s.m.s. with its associated O"-algebra of Borel sets, (f!t, ff) an arbitrary measurable space, and n a probability measure on the product space (.:'!', Jlt'). Then there exists a family Q(Bix) such that (i) Q( ·lx) is a probability measure on~ for each fixed x E :?l"; (ii) Q(BI·) is an ff-measurable function on;?[ for each fixed BE~; (iii) (Al.5.1) is satisfied for all A E ff, BE~- For details of the proof of this proposition, together with variants of it, we refer to the exercises in Chapter 6. We consider finally the product of a general family of measurable spaces, say {(.ot;, ff,): t E .'1}, where .'1 is an arbitrary (finite, countable, or uncount• able) indexing set. Once again the cylinder sets play a basic role. A measurable cylinder set in fl' = Xt e :r (f!tt) is a set of the form C(t1 , ... , tN; B 1 , ••• , BN) = {x(t): x(t;) E B;, i = 1, ... , N}, where B; E ff., is measurable for each i = 1, ... , N. Such sets form a semiring; their finite disjoint unions form a ring; and the generated O"-ring we denote by ffoo = Q9 ff,. te:T This construction can be used to define a product measure on ff00 , but greater interest centres on the extension problem: given a system of measures n(al defined on finite subfamilies 9fal = ff., (8) ff, 2 (8) · · · ® ff,N, where (O") = {t 1 , ..• , tN} is a finite selection of indices from .'1, when can they be extended to a measure on $'00 ? It follows from the extension theorem Al.3.111 that the necessary and sufficient condition for this to be possible is that the given measures must give rise to a countably additive set function on the ring generated by the measurable cylinder sets. As with the previous result, count• able additivity cannot be established without some additional assumptions; again it is convenient to put these in topological form by requiring each of the f!tt to be a c.s.m.s. Countable additivity then follows by a variant of the usual compactness argument, and the only remaining requirement is that the 606 APPENDIX 1. A Review of Some Basic Concepts given measures should satisfy the obviously necessary consistency conditions stated in the theorem below. Theorem A1.5.IV (Kolmogorov Extension Theorem). Let !T be any arbitrary index set, and for t E !T suppose (,q[1, ff,) is a c.s.m.s. with its associated Borel a-algebra. Suppose further that for each finite subfamily (a)= {t 1 , ... , tN} of indices from !T, there is given a probability measure n on ~a> = ff,, ® · · · ® ff,N. In order that there exist a measure non !i'00 such that for all (a), n is the projection of n onto ~a>• it is necessary and sufficient that for all (a), (ad, (a2 ), (i) n depends only on the choice of indices in (a), not on the order in which they are written down; and (ii) if (ad s;;; (a2 ), then n Written out more explicitly in terms of distribution functions, condition (i) becomes (in an obvious notation) the condition of invariance under simultane• ous permutations: if p 1 , ... , PN is a permutation of the integers 1, ... , N, then F.(N) (X X ) = F.(N) (X X ) tt,_ .. ,tN 1, ... , N tPt'"""''PN Pt'"""' PN" Similarly, condition (ii) becomes the condition of consistency of marginal distributions, namely, that Fi~~.~~N·'•·····•k(xl, ... , xN, oo, ... , oo) = Ft~ ... tN(xl, ... , xN). The measure n induced on ff'00 by the fidi distributions is called their projective limit. Clearly, if stochastic processes have the same fidi distribu• tions, they must also have the same projective limit. Such processes may be described as being equivalent or versions one of the other. Theorem Al.5.1V is discussed in a slightly more general form in Partha• sarathy (1967, §§5.1-5.5) to which reference may be made for proof and further detail. APPENDIX 2 Measures on Metric Spaces A2.1. Borel Sets, Dissecting Systems, and Atomic and Diffuse Measures If (q", 0/1) is a topological space, the smallest a-algebra containing the open sets is called the Borel a-algebra. Iff: q-- ~is any real-valued continuous function, then the set {x:f(x) < c} is open in 0/1 and hence measurable. It follows that f is measurable. Thus, every continuous function is measurable with respect to the Borel a-algebra. It is necessary to clarify the relation between the Borel sets and various other candidates for useful a-algebras that suggest themselves, such as (a) the Baire sets, belonging to the smallest a-field with respect to which the continuous functions are measurable; (b) the Borelian sets, generated by the compact sets in q"; and (c) if q- is a metric space, the a-algebra generated by the open spheres. We show that, with a minor reservation concerning (b), all four concepts coincide when q" is a c.s.m.s. More precisely, we have the following result. Proposition A2.1.1. Let q" be a metric space and 0/1 the topology induced by the metric. Then (i) the Baire sets and the Borel sets coincide; (ii) if q" is separable, then the Borel a-algebra is the smallest a-algebra contain• ing the open spheres; (iii) a Borel set is Borelian if and only if it is a-compact, that is, if it can be 608 APPENDIX 2. Measures on Metric Spaces covered by a countable union of compact sets. In particular, the Borel sets and the Borelian sets coincide if and only if the whole space is a-compact. The proof of (i) depends on the lemma below, of importance in its own right; (ii) depends on the fact that when !!£ is separable, every open set can be represented as a countable union of open spheres; (iii) follows from the fact that all closed subsets of a compact set are compact and hence Borelian. Lemma A2.1.11. Let F be a closed set in the metric space !!£, U an open set containing F, and IF(·) the indicator function of F. Then there exists a sequence of continuous functions Un(x)} such that (i) 0 s; fn(x) s; 1 (x E !!£); (ii) f,(x) = 0 outside U; (iii) f,(x)! IF(x) as n--+ oo. PROOF. Let f,(x) = p(x, Uc)/[p(x, Uc) + 2n p(x, F)], where for any set C p(x, C) = inf p(x, y) y€C Then the sequence f,(x) has the required properties. D It is clear that in a separable metric space the Borel sets are countably generated. The next lemma exhibits a simple example of a countable semiring of open sets generating the Borel sets. Lemma A2.1.111. Let !!£ be a. c.s.m.s., !!) a countable dense set in !!£, and 9'0 the class of all finite intersections of open spheres S(d, r) with centres d E !!) and rational radii. Then (i) 90 and the ring d 0 generated by 90 are countable; and (ii) 9'0 generates the Borel a-algebra in !!£. It is also a property of the Borel sets in a separable metric space, and of considerable importance in the analysis of sample-path properties of point processes, that they include a dissecting system defined as follows. Definition A2.l.IV. The sequence !I= {ff,} of finite partitions ff, = {Ani: i = 1, ... , kn} (n = 1, 2, ... ) consisting of Borel sets in the space !!£ is a dissecting system for !!£ when, in addition to the partition properties that Ani n Ani = 0 for i #- j and An 1 u · · · u Ank" = !!£, the sequences are nested (meaning that An-l,i n Ani = Ani or 0) and separate points of!!£ (meaning that for any given distinct points x, y E !!£, there exists an integer n such that x E Ani implies y ¢ An;). Proposition A2.l.V. Every separable metric space!!£ contains a dissecting system. A2.1. Borel Sets, Dissecting Systems, and Atomic and Diffuse Measures 609 PROOF. Let {d 1 , d2 , •••-} = !?) be a separability set for!!£ (i.e., f2 is a countable dense set in !!£). Take any pair of distinct points x, y e !!£; their distance apart equals (J =p(x, y) > 0. We can then find dm, dn in f2 such that p(dm, x) < (Jj2, p(dn, y) < (Jj2, and so the spheres S(dm, (Jj2), S(dn, (Jj2), which are Borel sets, certainly separate x and y. We have essentially to embed such separating spheres into a sequence of sets covering the whole space. For the next part of the proof it is convenient to identify one particular element in each fJ, (or it may possibly be a null set for all n sufficiently large) as An 0 : this entails no loss of generality. Definite the initial partition {Au} by Au= S(d1 , 1), A 10 =!!£\Au. Observe that!!£ is covered by the countably infinite sequence {S(dn, 1)}, so the sequence of sets {A~ 0 } defined by A~ 0 = !!£\ (Q S(d., 1)) converges to the null set. For n = 2, 3, ... and i = 1, ... , n, define Bn; = S(d;, 2- p(x, z) < 2, 2-(m+r- j) < (J = p(x, y), for any z E cm+r,j• and therefore the partition {cm+r,;}, and hence also {Am+r.J, separates x andy. D Trivially, if f7 is a dissecting system for!!£, the nonempty sets of f7 11 A (in an obvious notation) constitute a dissecting system for any A e ?-~!¥". If A is also compact, the construction of a dissecting system for A is simplified by applying the Heine-Borel theorem to extract a finite covering of A from the countable covering {S(d., 2-n): r = 1, 2, ... }. Definition A2.1.VI. The ring of sets generated by finitely many intersections and unions of elements of a dissecting system is a dissecting ring. 610 APPENDIX 2. Measures on Metric Spaces The chief motivation for introducing the concept of a dissecting system is that it facilitates the discussion of atomic and nonatomic components of a measure J.1. (i.e., nonnegative a-finite countably additive set function) on a metric space fl£. Call x E fl£ an atom of J.1. if JJ.( {x}) > 0, and call the measure bx defined on Borel sets A by Jx(A) = 1 if x E A, = 0 otherwise, Dirac measure (at x). A measure with only atoms is purely atomic, while a measure with no atoms is diffuse. Given a dissecting system:!/ for fl£, there is a well-defined nested sequence {T,(x)} c :!/such that 00 n T,(x) = {X}, SO JJ.(T,(x))---+ J.l.( {X}) for n---+ 00, n=l and it follows that xis an atom of J.1. if and only if JJ.(T,(x)) ~ e (all n) for some e > 0; indeed, any e in 0 < e:::;; JJ.({x}) will do. Given e > 0, we can identify all atoms of J.1. of mass JJ.( {x}) ~ e, and then using a sequence {ei} with ei t 0 asj---+ oo, all atoms of J.1. can be identified. Note that, because J.1. is a-finite, it can have at most countably many atoms, so identifying them as {X/ j = 1, 2, ... } say, and writing bi = JJ.( {xi}), the measure which clearly consists only of atoms, is the atomic component of the measure J.l.. The measure 00 J.1.a{·) =JJ.(·)- L bjbxJ) j=l has no atoms and is the diffuse component of J.l.. It is thus clear that any measure J.1. as above has a unique decomposition into atomic and diffuse components. [For further details, see Kallenberg (1975, pp. 10-11).] A2.2. Regular and Tight Measures In this section we examine the extent to which the values of a finitely or countably generated set function defined on some class of sets can be approxi• mated by their values on either closed or compact sets. Definition A2.2.1. (i) A finite or countably additive, nonnegative set function J.1. defined on the Borel sets is regular if, given any Borel set A and e > 0, there exist open and closed sets G and F, respectively, such that F s A s G and JJ.( G - A) < e and JJ.(A - F) < e. A2.2. Regular and Tight Measures 611 (ii) It is compact regular if, given any Borel set A and e > 0, there exists a compact set C such that C ~ A and J.L(A - C) < e. We first establish the following. Proposition A2.2.11. If f!{ is a metric space, then all totally finite measures on 86!!f are regular. PROOF. Let J.l be a totally finite, additive, nonnegative set function defined on 86!!f. Call any A E 86!!f J.L-regular if it can be approximated by its values on open and closed sets in the manner of Definition A2.2.1. The class of J.L-regular sets is obviously closed under complementation. It then follows from the inclusion relations (A2.2.1a) and (A2.2.1b) that the class is an algebra if J.l is finitely additive and a cr-algebra if J.l is countably additive. [In the latter case, the countable union U.F.. in (A2.2.1a) may not be closed, but we can approximate J.L(U .. F.. ) by J.L(Uf= 1 F.. ) to obtain a set that is closed and has the required properties; similarly, in (A2.2.lb) we can approximate J.L(n,. G,.) by J.L(nf= 1 G.. JJ Moreover, if J.l is cr-additive, the class also contains all closed sets, for ifF is closed, the halo sets (A2.2.2) F" = U S(x, e)= {x: p(x, F)< e} xeF form, for a sequence of values of e tending to zero, a family of open sets with the property F"! F; hence, it follows from the continuity Lemma A1.3.11 that J.L(F") -+ J.L(F). In summary, if J.l is countably additive the J.L-regular sets form a cr-algebra containing the closed sets, and therefore the class must coincide with the Borel sets themselves. D Note that this proof does not require either completeness or separability. Compact regularity is a corollary of this result and the notion of a tight measure. Definition A2.2.111. A finite or countably additive set function J.l is tight if, given e > 0, there exists a compact set K such that J.l(f!£ - K) is defined and J.l(f!£ - K) < e. Lemma A2.2.IV. If f£ is a complete metric space, a Borel measure is compact regular if and only if it is tight. 612 APPENDIX 2. Measures on Metric Spaces PRooF. Given any Borel set A it follows from Proposition A2.2.11 that there exists a closed set e £; A with Jl(A - C) < ej2. If Jl is tight, choose K so that Jl(?l" - K) < e/2. Then the set en K is a closed subset of the compact set K and hence is itself compact; it also satisfies Jl(A - e 11 K) :::;; Jl(A - C) + Jl(A - K) < e, which establishes the compact regularity of Jl. If, conversely, Jl is compact regular, tightness follows on taking ?£ = K. D Proposition A2.2.V. If?£ is c.s.m.s., every Borel measure Jl is tight and hence compact regular. PRooF. Let!!) be a separability set for?£; then for fixed n, U S(d, n-1) = ?£, de~ and so by the continuity lemma, we can find a finite set d1 , ... , dk k(n) ) Jl ( ?£ - i~ S(d;, n-1) < e/2n. Now consider K = 0[Q S(d;, n-1 )]. It is not difficult to see that K is closed and totally bounded, and hence com• pact, by Proposition A1.2.11 and that Jl(?l" - K) < e. Hence, Jl is tight. D The above results establish compact regularity as a necessary condition for a finitely additive set function to be countably additive. The next proposition asserts its sufficiency. The method of proof provides a pattern that is used with minor variations at several important points in the further development of the theory. Proposition A2.2.VI. Let d be a ring of sets from the c.s.m.s. ?£and Jl a finitely additive, nonnegative set function defined and finite on d. A sufficient condition for Jl to be countably additive on d is that, for every A E d and e > 0, there exists a compact set e £; d such that Jl(A - C) < e. PROOF. Let {An} be a decreasing sequence of sets in d with An~ 0; to establish countable additivity for Jl it is enough to show that Jl(An) --.. 0 for every such sequence. Suppose to the contrary that Jl(An) ~ ~ > 0. By assumption, there exists for each n a compact set en for which en £; An and Jl(An - en) < ~/2n+t. By (A2.2.1), A2.2. Regular and Tight Measures 613 Since d is a ring, every finite union U~=l (Ak - Ck) is an element of d, and so from the finite additivity of p, p, (An - k61 Ck) ~ kt1 rx/2k+l < rx/2. Thus, the intersection nk=l Ck is nonempty for each n, and it follows from the finite intersection part of Proposition A1.2.11 that nk'= 1 Ck is nonempty. This gives us the required contradiction to the assumption An! 0. D Corollary A2.2.VII. A finite, finitely additive, nonnegative set function defined on the Borel sets off£ is countably additive if and only if it is compact regular. We now prove an extension of Proposition A2.2.VI, which plays on impor• tant role in developing the existence theorems of Chapter 6. It is based on the notion of a self-approximating ring and is a generalization of the concept of a covering ring given in Kallenberg (1975). Definition A2.2.VIII. A ring d of sets of the c.s.m.s. f!( is a self-approximating ring if, for every A e d and e > 0, there exists a sequence of closed sets {Fk(A; e)} such that (i) Fk(A; e) Ed (k = 1, 2, ... ); (ii) each set Fk(A; e) is a contained within a sphere of radius e; (iii) U1 Fk(A; e) = A. Kallenberg was concerned with the case when f£ is locally compact, in which case it is possible to require the covering to be finite, so that the lemma below effectively reduces to Proposition A2.2.VI. The general version is based on an argument in Harris (1968). The point is that it allows checking for countable additivity to be reduced to a denumerable set of conditions. Lemma A2.2.IX. Let d be a self-approximating ring of subsets of the c.s.m.s. f!( and Jl a finitely additive, nonnegative set function defined on d. In order that Jl have an extension as a measure on u(A) it is necessary and sufficient that for each A e d, (A2.2.3) !~ p,(Q Fi(A; e))= p,(A), where the notation of Definition A2.2.VIII is to be understood. PROOF. Necessity follows from the continuity lemma. We establish sufficiency by contradiction: suppose that p, is finitely additive and satisfies (A2.2.3) but that p, cannot be extended to a measure on a(d). From the continuity lemma again it follows that there exists rx > 0 and a sequence of sets An e d, with An! 0, such that (A2.2.4) p,(An) ~ rx. 614 APPENDIX 2. Measures on Metric Spaces For each k, use (A2.2.3) to choose a set mk Fk = U F;(Ak; 1/k) i=l that is closed, can be covered by a finite number of k-1 spheres, and satisfies J.t(Ak - Fk) s rx./2k+ 1• From (A2.2.1) we have ( Ak - 0jj) s; y(Ai - fj), which, using the additivity of J.t, implies that J.t ( 0}j) ~ rx./2 > 0. Thus, the sets jj have the finite intersection property. To show that their complete intersection is nonempty, choose any xk e n~ jj. Since F1 can be covered by a finite number of 1-spheres, there exists a subsequence {x~} that is wholly contained within a sphere of radius 1. Turning to F2 , we can select a further subsequence x;, which for k ~ 2 lies wholly within a sphere of radius!. Proceeding in this way by induction, we finally obtain by a diagonal selection argument a subsequence {xkJ such that for j ~ j 0 all terms are contained within a sphere of radius 1/j0 . This is enough to show that {xk.} is a Cauchy sequence that, since!!( is complete, has a limit point X, say. For each k the xj are inn~ Fn for all sufficiently largej. Since the sets are closed this implies x e Fk for every k. But this implies also that x e Ak and hence X E nf Ak, which contradicts the assumption An! 0. The contra• diction shows that (A2.2.4) cannot hold, and so completes the proof of the lemma. 0 Let us observe finally that self-approximating rings do exist. A standard example, which is denumerable and generating as well as self-approximating, is the ring a countable family of closed sets satisfying (iii) of Definition A2.2.VIII for the given closed sphere. It is obvious that ~ is closed under finite unions, and that, from the relation CQ Fj) n (91 F~) = iQ kQl (Fj n FD, ~is also closed under finite intersections. Since~ contains all closed spheres and also their complements (which are open), ~ contains rc. Thus every set in rc can be approximated by closed spheres in rc, and so rc is self-approximating as required. A2.3. Weak Convergence of Measures We make reference to the following notions of convergence of a sequence of measures on a metric space. Definition A2.3.1. Let {.un: n ;;?: 1}, ,u, be totally finite measures in the metric space f£. Then (i) .Un --+ .u weakly if Jf d.un --+ Jf d.u for all bounded continuous functions f on:£; (ii) .Un --+ .u vaguely if Jf d.un --+ Jf d.u for all bounded continuous functions f on :£, which vanish outside a compact set; (iii) .Un--+ .U strongly (or in variation norm) if II.Un- .ull --+ 0. The last definition corresponds to strong convergence in the Banach space of all totally finite signed measures on :£, for which the total variation constitutes a genuine norm. The first definition does not correspond exactly to weak convergence in the Banach space sense, but it reduces to weak star (weak*) convergence when f£ is compact (say, the unit interval) and the space of signed measures on f£ can be identified with the adjoint space to the space of all bounded continuous functions on :£. Vague convergence is particularly useful in the discussion of locally compact spaces; in our discussion a some• what analogous role is played by the notion of weak hat convergence (i.e., w-convergence) to be introduced in Section A2.6: it is equivalent to vague convergence when the space is locally compact. Undoubtedly, the central concept for our purposes is the concept of weak convergence. Not only does it lead to a convenient and internally consistent topologization of the space of realizations of a random measure, but it also provides an appropriate framework for discussing the convergence of random measures conceived as probability distributions on this space of realizations. In this section we give a brief treatment of some basic properties of weak convergence, following closely the discussion in Billingsley (1968) to which we refer for further details. 616 APPENDIX 2. Measures on Metric Spaces Theorem A2.3.11. Let :1£ be a metric space, {.Un• n 2: 1}, ,u, measures on PAP£. Then the following statements are equivalent: (i) .Un--+ ,u weakly; (ii) .Un(fl£) --+ ,u(fl£) and lim SUPn-oo .Un(F) ~ ,u(F) for all closed F E PAP£; (iii) .Un(fl£)--+ ,u(fl£) and lim infn-oo .Un(G) 2: ,u(G) for all open G E PAP£; (iv) .Un(A)--+ ,u(A) for all Borel sets A with ,u(oA) = 0 (i.e., all ,u-continuity sets). PROOF. We shall show that (i) = (ii)-=- (iii)= (iv) = (i). Given a closed set F, choose any fixed v > 0 and construct a [0, 1]-valued continuous function f that equals 1 on F and vanishes outside pv [see (A2.2.2) and Lemma A2.1.11]. We have for each n 2: 1 .Un(F) ~ If d,un ~ .Un(F'); hence, if (i) holds, But P l F as v l 0, and by the continuity lemma we can choose v so that, given any B > 0, ,u(P) ~ ,u(F) + B. Since B is arbitrary, the second statement in (ii) follows, while the first is trivial if we take f = 1. Taking complements shows that (ii) and (iii) are equivalent. When A is a ,u-continuity set, ,u(A 0 ) = ,u(A), and so supposing (iii) holds, and hence (ii) also, we have on applying (ii) to A and (iii) to A 0 that 0 0 lim sup .Un(A) ~ lim sup .Un(A) ~ ,u(A) = ,u(A ) ~ lim inf .Un(A ) ~ lim inf .Un(A). Thus, equality holds throughout and .Un(A)--+ ,u(A), so that (iv) holds. Finally, suppose that (iv) holds. Let f be any bounded continuous function on :1£, and let the bounded interval [o:', o:"] be such that o:' < f(x) < o:" for all x E !1£. Call o: E [o:', o:"] a regular value off if ,u{x:f(x) = o:} = 0. At most a countable number of values of o: can be irregular, while for any o:, {3 that are regular values, {x: o: < f(x) ~ {3} is a ,u-continuity set. From the boundedness off on :1£, given any B > 0, we can partition [o:', o:"] by a finite set of points o: 0 = o:', ... , o:N = o:" with o:; < o:i+ 1 ~ o:; + B fori= 0, ... , N- 1, and from the countability of the set of irregular points (if any), we can moreover assume that these o:; are all regular points off Defining A; = { x: o:;_ 1 < f(x) ~ o:;} for i = 1, ... , N, and then N N fL(x) = L 0:;-1 IA,(x), fu(x) = L o:JA,(x), i=1 i=1 each A; is a ,u-continuity set, fL(x) ~ f(x) ~ fu(x), and by (iv), A2.3. Weak Convergence of Measures 617 ffLdJl = i~ CXi-lJl(Ai) = !~~ i~ CXi-lJln(Ai) = !~~ ffLdJln ~ lim ffudJln = ffudJl, n-+oo the extreme terms here differing by at most eJl(!!(). Since e is arbitrary and JfLdJln ~ Jf dJln ~ JfudJln, it follows that we must have Jf dJln--+ Jf djl for all bounded continuous f, that is, Jln--+ Jl weakly. D Since the functions used in the proof that (i) implies (ii) are uniformly continuous, we can extract from the proof the following useful condition for weak convergence. Corollary A2.3.III. Jln--+ Jl weakly if and only if Jf dJln--+ Jf djl for all bounded and uniformly continuous functions f: !!(--+ ~- A class CC of sets with the property that (A2.3.1) Jln(C)--+ Jl(C) (all C E CC) implies Jln--+ Jl weakly, is called by Billingsley a convergence-determining class. Adapting this ter• minology, (iv) of the theorem above asserts that the Jl-continuity sets form a convergence-determining class. Any convergence-determining class is neces• sarily a determining class, but the converse need not be true. In particular circumstances, it may be of considerable importance to find a convergence• determining class that is smaller than the classes in Theorem A2.3.11. While such classes often have to be constructed to take advantage of particular features of the metric space in question, the general result below is also of value. In this proposition a covering semiring is a semiring with the property that every open set can be represented as a finite or countable union of sets from the semiring. If!!( is separable, an important example of such a semiring is obtained by first taking the open spheres S(dk, ri) with centres at the points {dk} of a countable dense set and radii {ri} forming a countable dense set in (0, 1), then forming finite intersections, and finally taking proper differences. Proposition A2.3.IV. Any covering semiring, together with the whole space!!(, forms a convergence-determining class. PRooF. Let G be an open set, so that by assumption we have 00 G = U Ci for some C; e !/', 1 where!/' is a generating semiring. Since the limit Jl in (A2.3.1) is a measure, given e > 0, we can choose a finite integer K such that Jl( G- iQ C;) ~ e/2; 618 APPENDIX 2. Measures on Metric Spaces that is, JJ.(G) ~ J1. (Q C;) + B/2. Furthermore, since CC is a semiring, Uf=1 C; can be represented as a finite union of disjoint sets in CC. From (A2.3.1) it therefore follows that there exists N such that for n;;:::; N Hence, we obtain JJ.(G) ~lim inf Jl.n (u c;) + e ~lim inf JJ.n(G) +e. n-+oo 1 n-+oo Since e is arbitrary, (iii) of Theorem A2.3.11 is satisfied, and therefore Jl.n -.. J1. weakly. D We investigate next the preservation of weak convergence under mappings from one metric space into another. Let !!f, I{IJ be two metric spaces with associated Borel a-algebras EJIP£, fJit!f, and f a measurable mapping from (!!f, EJIP£) into (W, 91'!1). Any measure J1. on EJIP£ induces a measure J-Lf- 1 on fJit!f, where for B E fJit!f, Now let Jl.n be any sequence of measures on EJIP£ such that Jl.n-.. J1. weakly. Does it follow that J-Lnf- 1 -.. J-Lf- 1 weakly? The answer is no in general; a sufficient condition is given in the next proposition, where D1 is the set of points of discontinuity of f [recall that f is continuous at x if Pf!l(f(x'),J(x))-.. 0 whenever PPE(x', x)-.. 0]. Proposition A2.3.V. Let (!!f, EJIP£), (W, 91'!1) be metric spaces and fa measurable mapping of (!!f, EJIP£) into (W, 91'!1 ). Suppose Jl.n -.. J1. weakly on !![ and J-L(D1 ) = 0; then Jl.nf- 1 -.. J-Lf- 1 weakly. PRooF. Let B be any Borel set in 91'!1 and x any point in the closure of f-1 (B). For any sequence of points xn Ej-1(B), such that Xn-.. x, either x E D1 or f(xn)-.. f(x), in which case x Ej-1(ii). Arguing similarly for the complement, we find that (A2.3.2) Now suppose Jl.n-.. J1. weakly on EJIP£, and consider the image measures Jl.nf-1, J-Lf- 1 on 91'!1. Let B be any continuity set for J-Lf- 1• It follows from (A2.3.2) and the assumption of the proposition that f-1 (B) is a continuity set for JJ.. Hence, for all such B, (J-Lnf- 1 )(B) = J.ln(J- 1 (B)) -.. J-L(f- 1 (B)) = (J-Lf- 1 )(B), that is, Jl.nf- 1 -.. J-L!- 1 weakly. D A2.4. Compactness Criteria for Weak Convergence 619 A2.4. Compactness Criteria for Weak Convergence In this section we call a set .A of totally finite Borel measures on f!l relatively compact for weak convergence if every sequence of measures in .A contains a weakly convergent subsequence. It is shown in Section A2.5 that weak convergence is equivalent to convergence with respect to a certain metric, and that if f!l is a c.s.m.s., the space of all totally finite Borel measures on f!l is itself a c.s.m.s. with respect to this metric. We can then appeal to Proposition Al.2.11 and conclude that a set of measures is compact (or relatively compact) if and only if it satisfies any of the criteria (i)-(iv) of that proposition. Our aim in this section is to establish the following criterion for compact• ness. Theorem A2.4.1 (Prohorov's Theorem). Let f!l be a c.s.m.s. Necessary and sufficient conditions for a set .A of totally finite Borel measures on f!l to be relatively compact for weak convergence are (i) the total masses Jl(f!l) are uniformly bounded for Jl E .A; and (ii) .A is uniformly tight-namely, given e > 0, there exists a compact K such that, for all Jl E .A, (A2.4.1) Jl(f!l - K) < e. PROOF. We first establish that the uniform tightness condition i.s necessary, putting it in the following alternative form. Lemma A2.4.11. A set .A of measures is uniformly tight if and only if, for all e > 0 and {J > 0, there exists a finite family of fJ-spheres (i.e., of radius J) sl' ... ' SN such that (A2.4.2) PROOF OF LEMMA. If the condition holds, we can find, for every k = 1, 2, ... , a finite union Ak of spheres of radius 1/k such that Jl(f!l - Ak) ::5; ej2k for all Jl E .A. Then the set 00 K= n Ak k=l is totally bounded and hence compact, and for every Jl E .A, 00 Jl(f!l - K) ::5; L Jl(f!l - Ak) < e. k=l Thus, .A is uniformly tight. Conversely, if .A is uniformly tight and, given e, we choose a compact K to satisfy (A2.4.1), then for any {J > 0, K can be covered by a finite set of J-spheres, and so (A2.4.2) holds. D 620 APPENDIX 2. Measures on Metric Spaces Returning now to the main theorem, suppose if possible that..,(( is relatively compact but (A2.4.2) fails for some e > 0 and () > 0. Since we assume f!l is separable we can write f!l = U1 Sk, where each Sk is a {)-sphere. On the other hand, for every finite n, we can find a measure J-ln E ..,(( such that (A2.4.3a) If in fact..,(( is relatively compact, there exists a subsequence J-ln. that converges J weakly to some limit J-t*. From (A2.4.3a) we obtain via (ii) of Theorem A2.3.11 that for all N > 0 J-t* (f!l - Usk) ~ lim sup J-lnj (f!l - Usk) ~ e. 1 ~~00 1 This contradicts the requirement that, because f!l- Uf Sk! 0. we must have J-t*(f!l- Uf Sk)-+ 0. Thus, the uniform tightness condition is necessary. As it is clear that no sequence {JJ.n} with J-ln(f!l)-+ oo can have a weakly convergent subsequence, condition (i) is necessary also. Turning to the converse, we again give a proof based on separability, although in fact the result is true without this restriction. Let us start by constructing a countable ring~ from the open spheres with rational radii and centres in a countable dense set, by taking first finite intersections and then proper differences, thus forming a semiring, and finally taking all finite disjoint unions of such differences. Now suppose that {JJ.n: n ~ 1} is any sequence of measures from ..,((_ We have to show that {JJ.n} contains a weakly convergent subsequence. For any A E ~. condition (i) implies that {JJ.n(A)} is a bounded sequence of real num• bers and therefore contains a convergent subsequence. Using a diagonal selection argument, we can proceed to extract subsequences {JJ.n) for which the J-ln .(A) approach a finite limit for each of the countable number of sets J A E ~-Let us write JJ.*(A) for the limit, and for brevity of notation set J-lni = J-t]. Thus, we have (A2.4.3b) J-t}(A)-+ J-t*(A) (all A E ~). This might seem enough to set up a proof, for it is easy to see that JJ.* inherits finite additivity from the J-t], and one might anticipate that the uniform tight• ness condition could be used to establish countable additivity. The difficulty is that we have no guarantee that the sets A E ~are continuity sets for J-t*, so that (A2.4.3b) cannot be relied on to give the correct value to the limit measure. To get over this difficulty, we have to develop a more elaborate argument incorporating the notion of a continuity set. For this purpose we introduce the class~ of Borel sets, which are JJ.*-regular in the following sense: given C E ~.we can find a sequence {An} of sets in~ and an associated sequence of open sets Gn such that An 2 Gn 2 C, and similarly a sequence of sets Bn E ~ and closed sets Fn with C 2 Fn 2 Bm the two sequences {An}, {Bn} having the property (A2.4.4) lim inf J-t*(An) = lim sup J-t*(Bn), = J-t(C), say. A2.4. Compactness Criteria for Weak Convergence 621 We establish the following properties of the class C(J. (1) C(J is a ring: Let C, C' be any two sets in C(J, and consider, for example, the difference C - C'. If {An}, {Gn}, {Bn}, {Fn} and {A~}, { G~}, {B~}, { F~} are the sequences for C and C', respectively, then An - B~ 2 Gn - F~ 2 C - C' 2 Fn - G~ 2 Bn - A~, with G - F~ open, Fn - G~ closed, and the outer sets elements of fJ1i since fJ1i is a ring. From the inclusion (An - B~) - (Bn - A~) ~ (An - Bn) u (A~ - B~}, we find that ll*(An - B~) and ll*(Bn - A~) have common limit values, which we take to be the value of Jl(C- C'). Thus, C(J is closed under differences, and similar arguments show that C(J is closed also under finite unions and intersections. (2) C(J is a covering ring: Let d be any element in the countable dense set used to construct fJii, and for rational values of r define h(r) = ll*(S(d, r)). Then h(r) is monotonic increasing, bounded above, and can be uniquely extended to a monotonic increasing function defined for all positive values of r and continuous at all save a countable set of values of r. It is clear that if r is any continuity point of h(r), the corresponding sphere S(d, r) belongs to C(J. Hence, for each d, we can find a sequence of spheres S(d, en) E C(J with radii en -+ 0. Since any open set in !!{ can be represented as a countable union of these spheres, C(J must be a covering class. (3) For every C E C(J, ll}(C)-+ Jl(C}: Indeed, with the usual notation we have ll*(An) =lim ll}(An) ~lim sup ll}(C) ~ liminf ll}(C) j-+oo j-+oo j-+oo ~ lim ll}(Bn} = ll*(Bn}. j-+oo Since the two extreme members can be made as close as we please to ll( C), the two inner members must coincide and equalJl(C}. (4) ll is finitely additive on C(J: This follows from (3) and the finite additivity of ll}· (5) If .A is uniformly tight, then ll is countably additive on C(J: Suppose that {Ck} is a sequence of sets from C(J, with Cd 0, but ll(Ck) ~a > 0. From the definition of C(J, we can find for each Ck a set Bk E fJ1i and a closed set Fk such that Ck 2 Fk 2 Bk and ll*(Bk) > /l(Ck)- aj2k+ 1; then lim inf Jl}(Fk) ~ lim ll}(Bk) = ll*(Bk) ~ a - aj2k+ 1• j-+oo j-+oo We then have /l( Cd = li':fl inf ll} (n Fn) + li~ sup ll} (ck - nFn) r-+oo 1 J-+OO 1 622 APPENDIX 2. Measures on Metric Spaces hence, for all k If now .A is uniformly tight, there exists a compact set K such that 11(!!t - K) < a/4 for all11 E .A. In particular, therefore, 11}( 0Fn) -11}( 0(FnnK)) < a/4 and so But this is enough to show that for each k, the sets (n~ Fn) n K are nonempty, and since (if!![ is complete) each is a closed subset of the compact set K, it follows from Theorem A1.2.II that their total intersection is nonempty. Since their total intersection is contained in nf en, this set is also nonempty, contradicting the assumption that en! 0- We can now complete the proof of the theorem without difficulty. From the countable additivity of 11 on<'{{, it follows that there is a unique extension of 11 to a measure on :JB?I· Since <'{{ is a covering class, and 11}( C)--+ 11( C) for C E <'{/, it follows from Proposition A2.3.III that 11} --+ 11 weakly, or in other words that the original sequence lln contains a weakly convergent subsequence, as required. D A2.5. Metric Properties of the Space A f£ Denote by .A?I the space of all totally finite measures on :JB?£, and consider the following candidate (the Prohorov distance) for a metric on .A?£, where F' is a halo set as at (A2.2.2): (A2.5.1) d(/1, v) = inf{e: e ;:::-: 0, and for all closed F s; f!{, 11(F) ~ v(F') + e and v(F) ~ 11(F') + e}. If d(/1, v) = 0, then 11(F) = v(F) for all closed F, so /1( ·)and v( ·)coincide. If d()., 11) = J and d(/1, v) = e, then ).(F) ~ 11(Ftl) + J ~ 11(Ftl) + J ~ v((Ftl)') + (J + e ~ v(Ftl+') + J + e, A2.5. Metric Properties of the Space .Itrr 623 with similar inequalities holding when A. and v are interchanged. Thus, the triangle inequality holds for d, showing that d is indeed a metric. The main objects of this section are to show that the topology generated by this metric coincides with the topology of weak convergence and to establish various properties of .Af1l' as a metric space in its own right. We start with an extension of Theorem A2.3.11. Proposition A2.5.1. Let f£ be a c.s.m.s. and vHf'£ the space of all totally finite measures on ;!If'£· Then each of the following families of sets in .Af1l' is a basis, and the topologies generated by these four bases coincide: (i) The sets {v: d(v, JL) < e} for all e > 0 and JL E .Af'l'. (ii) The sets {v: v(F;) < JL(F;) + e for i = 1, ... , k, lv(El")- JL(El")l < e} for all e > 0, finite families of closed sets F1, •.. , Fk, and J1. E .Afll'· (iii) The sets {v: v(G;) > JL(G;)- e fori= 1, ... , k, lv(El")- JL(El")l < e} for all e > 0, finite families of open sets G1, .•. , Gk, and JL E .Af'l'. (iv) The sets {v: Iv(A;) - JL(A;)I < e fori = 1, ... , k, Iv(El") - JL(El")l < e} for all e > 0, finite families of Borel sets A 1, ... , Ak with JL(oA;) = 0, and JLEvftfll'. PROOF. Each of the four families is specified as a family of neighbourhoods of an element JL E .Af'l'. Each family is closed under intersections and so satisfies the defining property of a basis. To show that the four bases are equivalent, it is enough to show that every neighbourhood of Jl. in one family contains a neighbourhood of JL in each of the others. Taking the families in the order (iv), (iii), (ii), (i), (iv), we show that each neighbourhood in one family contains a neighbourhood of the next family. Suppose there is given a neighbourhood of J1. as at (iv). Take as open set G; the interiors Ar of A;, together with the complements (~)< of their closures, and consider any measure v in the neighbourhood as at (iii) specified by v(Ar} > JL(Ar}- e/2, v((~)<) > JL((~)<)- e/2, and lv(El")- JL(El")l < ej2. Then since JL(oA;) = 0, it follows that for each i we have v(A;) ~ v(Ar} > JL(Ar} - e/2 > JL(A;) - e and JL(A;) = JL(~) = JL(f£) - JL((~)<) > v(El") - v((~)<) - e/2 - e/2 = v(~) - e ~ v(A;) - e, showing that v lies within the given neighbourhood of JL. Suppose next there is given a neighbourhood as at (iii); set F; = Gf, and consider any measure v in the neighbourhood as at (ii) with e/2 in place of e. Then v(G;) = v(El")- v(Gf) > JL(El")- JL(Gf)- e/2- e/2 = JL(G;)- e, showing that v lies within the given neighbourhood of JL. Given a neighbourhood as at (ii), the F; being closed, we can find a (J in 0 < (J < e/2 for which, for i = 1, ... , k, JL(Fl) < JL(F;) + e/2. 624 APPENDIX 2. Measures on Metric Spaces Consider the sphere S(J.1, b) with centre J.1 and radius b in the metric d. For any v E S(J.t, b), v(F;) < J.l(F;~) + b < J.l(F;) + e/2 + e/2 = J.l(F;) + B, so that v also lies in the given neighbourhood (i). Finally, suppose there is given a neighbourhood as at (i). Use the separa• bility of:!{ to cover:!{ with a countable number of spheres S1 , S2 , ... of radius e/3 or less. For any given x E f!£, the quantities f.l(S(x, r)) are monotonic increasing in r, and therefore continuous in r except for a countable set of values; so we can choose r < e/3 such that S(x, r) is a continuity set for f.l· We can therefore suppose without loss of generality that the S; are also J.l• continuity sets. Since f.1 (Q s) i f.l(f!£) (N--+ oo ), we can choose N so that f.1(Uf= 1 S;) > f.l(f!£) - e/3. Now consider any v in the neighbourhood as at (iv) specified by where the finite number of sets A; consists of the spheres S1 , ... , SN, all possible finite unions of these spheres, and the set f!l"\ Uf= 1 S;, noting that all these sets are J.l-continuity sets. For any closed set F, denote by F' the union of those spheres S; that intersect F, so that F ~ F' u ( f!£\Q s) and F' ~ F', and for v in the neighbourhood as described, v(F) ~ v(F') + v ( f!£\Q S;) < J.l(F') + f.1 ( f!£\Q s) + e/3 + e/3 < J.l(F') + B, and J.l(F) ~ J.l(F') + J.1 ( f!£\Q S;) < v(F') + e/3 + e/3 < v(F') +e. Thus, v E S(f.l, e), completing the proof of the proposition. 0 The weak convergence of J.ln to J.1 is equivalent by Theorem A2.3.11 to J.ln --+ J.l in each of the topologies (ii), (iii), and (iv), and hence by the proposition to d(J.ln, J.l)--+ 0. The converse holds, so we have the following. Corollary A2.5.11. For J.ln, J.1 E A?£, J.ln--+ J.1 weakly if and only if d(f.ln, J.l)--+ 0. Having established the fact that the weak topology is a metric topology, it makes sense to ask whether A?£ is separable or complete with this topology. A2.5. Metric Properties of the Space .Af£ 625 Proposition A2.5.m. If!!{ is a c.s.m.s. and .It:'£ is given the topology of weak convergence, then .ItP£ is also a c.s.m.s. PROOF. We first establish completeness by using the compactness criteria of the preceding section. Let {J.Ln} be a Cauchy sequence in .It:'£; we show that it is uniformly tight. Let positive e and ~ be given, and choose positive '1 < min(e/3, ~/2). From the Cauchy property there is an N for which d(J.Ln, J.LN) < '1 for n ~ N. Since J.LN itself is tight,!!{ can be covered by a sequence of spheres St, S2 , ••• of radius '1 and there is a finite K for which J.LN(!!£) - J.lN (Q S;) < '1· For n > N, since d(J.Lm J.LN) < '7, J.Ln(!!£) - J.LN(!!£) < '1 and so that < J.Ln(!!£) - J.lN (Q s) + '7 ~ IJ.Ln(!!£) - J.LN(!!£)1 + IJ.LN(!!£) - J.LN (Q s) I + '1 ~ 3'1 < B. It follows that for every e and ~ we can find a finite family of~ spheres whose union has J.Ln measure within e of J.Ln(!!£), uniformly inn. Hence, the sequence {J.Ln} is uniformly tight by Lemma A2.4.11 and relatively compact by Theorem A2.4.1 [since it is clear that the quantities J.Ln(f£) are bounded when {J.Ln} is a Cauchy sequence]. Thus, there exists a limit measure such that J.Ln-+ J.L weakly, which implies by Corollary A2.5.11 that d(J.Ln, J.L) -+ 0. Separability is rather easier to establish, as a suitable dense set is ready to hand in the form of the measures with finite support, that is, those that are purely atomic with only a finite set of atoms. Restricting the atoms to the points of a separability set !!} for!!{ and their masses to rational numbers, we obtain a countable family of measures,!!}' say, which we now show to be dense in .It:'£ by proving that any sphere S'(J.L, e)~ .It:'£ contains an element of!!}'. To this end, first choose a compact set K such that J.L(!!£\K) < ej2, which is possible because J.L is tight. Now cover K with a finite family of disjoint sets At, ... , A", each with nonempty interior and radius e or less. [One way of constructing such a covering is as follows. First cover K with a finite family of open spheres, St, ... , Sm say, each ofradius e. Take At = St, A 2 = S2 n A'l, A3 = S3 n (At u A2 )c, and so on, retaining only the nonempty sets in this construction. Then S2 n A'l is open, and either empty, in which case S2 ~At 626 APPENDIX 2. Measures on Metric Spaces so S2 £ A 1 and A 2 is empty, or else has nonempty interior. It is evident that each ,A; has radius e or less and that they are disjoint.] For each i, since A; has nonempty interior, we can choose an element X; of the separability set for f!£ with X; E A;; give X; rational mass J.l; such that J.t(A;) ~ J.l; ~ J.t(A;) - e/2N, and let J.l' denote a purely atomic measure with atoms at X; of mass J.l;· Then for an arbitrary closed set F, with L:' denoting Li:x,eF• J.t'(F) = L' J.l; ::;; L' J.t(A;) < J.t(F") + e, where we have used the fact that Ui:x,eFA; £ F" because A; has radius at most e. Furthermore, J.t(F} < J.t(K n F) + e/2 ::;; L:" J.t(F n A;) + e/2, where L:" denotes Li:A,nF?'0• so J.t(F) ::;; L:" J.l'(A;) + e/2 + e/2 ::;; J.t(F") + e. Consequently, d(J.l, J.t') < e, or equivalently, J.l' E S'(J.l, e) as required. D Denote the Borel a-algebra on Jtff by .?l(Jtff), so that from the results just established it is the smallest a-algebra containing any of the four bases listed • in Proposition A2.5.1. We use this fact to characterize .?l(Jtff). Proposition A2.5.IV. Let Y be a semiring generating the Borel sets .?Iff of f!£. Then .?l(Jtff) is the smallest a-algebra of subsets of Jtff with respect to which the mappings PROOF. Start by considering the class ~ of subsets A of f!£ for which {v: A2.6. Boundedly Finite Measures and the Space JlPI For applications to random measures, we need to consider not only totally finite measures on !IJ:r but also a-finite measures with the strong local finiteness condition contained in the following definition. Definition A2.6.1. A Borel measure J.1 on the c.s.m.s. !![ is boundedly finite if JJ(A) < oo for every bounded Borel set A. We write Ax for the space of boundedly finite Borel measures on !!l', and generally use the ~ notation for concepts taken over from finite to boundedly finite measures. The object of this section is to extend to A :r the results previously obtained for vi!:r: while most of these extensions are routine, they are given here for the sake of completeness. Consider first the extension of the concept of weak convergence. Taking a fixed origin x 0 E !!l', let S(r) = S(r, x0 ) for 0 < r < oo, and introduce a distance function d on A :r by setting (A2.6.1) d(JJ, v) =I"' e-rdr(J.l(r). v Proposition A2.6.II. Let {J.Lk: k = 1, 2, ... } and J.L be measures in Jl!'E; then the following conditions are equivalent: (i) J.Lk -+,;, .u; (ii) J!'E f(x)J.Lk(dx)-+ J!'E f(x).u(dx) for all bounded continuous functions f( ·) on f!£ vanishing outside a bouncfed set; (iii) there exists a sequence of spheres s PROOF. We show that (i) =(iii)= (ii) = (iv) = (i). Write the integral at (A2.6.1) for the measures J.lk and J.L as a(.uk. J.L) = IX) e_, gk(r)dr. so that for each k, gk(r) increases with rand is bounded above by 1. Thus, there exists a subsequence {kn} and a limit function g( ·) such that gk)r)-+ g(r) at all continuity points of g [this is just a version of the compactness criterion for vague convergence on ~: regard each gk(r) as the distribution function of a probability measure, so that there exists a vaguely convergent subsequence; see Corollary A2.6.V or any standard proof of the Helly-Bray results]. By dominated convergence, J!f e-' g(r)dr = 0 and hence, since g( ·)is monotonic, g(r) = 0 for all finite r > 0. This being true for all convergent subsequences, it follows that gk(r) -+ 0 for such r, and thus, for these r, d,(.ut>, .u<'>)-+ 0 ask-+ oo. In particular, this is true for an increasing sequence of values rn, corresponding to spheres {S(rn)} ={ Sn} say, on which therefore .utn> -+ .u<'n> weakly. Thus, (i) implies (iii). Suppose next that (iii) holds and that f is bounded, continuous, and vanishes outside some bounded set. Then the support off is contained in some S(r), and hence Jf d.ut> -+ Jf d.u<'>, which is equivalent to (ii). When (ii) holds, the argument used to establish (iv) of Theorem A2.3.11 shows that .Uk(C)-+ .u(C) whenever Cis a bounded Borel set with .u(oC) = 0. A2.6. Boundedly Finite Measures and the Space .IIf!l' 629 Finally, if(iv) holds and S(r) is any sphere that is a continuity set for Jl., then by the same theorem Jl.t>-+ Jl. While the distance function aof Definition A2.6.1 depends on the centre Xo of the family {S(r)} of spheres used there, the w-topology does not depend on the choice of x 0 . To see this, let {S~} be any sequence of spheres expanding to ?£,so that to any S~ we can first find n' for which S~ s; S(rn,), and then find n" for which S(rn,) s; S~~~. Now weak convergence within a given sphere is sub• sumed by weak convergence in a larger sphere containing it, from which the asserted equivalence follows. It should also be noted that for locally compact?£, w-convergence coincides with vague convergence. The next theorem extends to w-convergence the results in Propositions A2.5.111 and A2.5.1V. Theorem A2.6.111. (i) .A!¥ with thew-topology is a c.s.m.s. (ii) The Borel a-algebra !JB(Jt!¥) is the smallest a-algebra with respect to which the mappings A: .A!¥-+ IR given by A(Jl.) = Jl.(A) are measurable for all sets A in a semiring Y' of bounded Borel sets generating !JBf£, and in particular for all bounded Borel sets A. PROOF. To prove separability, recall first that the measures with rational masses on finite support in a separability set f!) for ?£ form a separability set f!)' for the totally finite measures on each Sn under the weak topology. Given e > 0, chooseR so that JR' e-'dr < e/2. For any J1. E .Af£, choose an atomic measure llR from the separability set for SR such that Jl.R has support in SR and dR(Jl.R, Jl. d,(Jl.~>, Jl.(rl) < e/2. Substitution in the expression for a shows that a(Jl.R, Jl.) < e, establishing that the union of separability sets is a separability set for measures in .A!¥· To show completeness, let {lld be a Cauchy sequence for d. Then each sequence of restrictions {Jl.t>} forms a Cauchy sequence ford, and so has a limit v, by Proposition A2.5.111. The sequence {v,} of measures so obtained is clearly consistent in the sense that v,(A) = v.(A) for s ~ r and Borel sets A of s•. Then the set function Jl.(A) = v,(A) 630 APPENDIX 2. Measures on Metric Spaces is uniquely defined on Borel sets A of S, and is nonnegative and countably additive on the restriction of .A[f[ to each S,. We now extend the definition of Jl to all Borel sets by setting Jl(A) = lim v,(A n S,), the sequence on the right being monotonic increasing and hence providing a limit (finite or infinite) for all A. It is then easily checked that Jl( ·) is finitely additive and continuous from below, and therefore countably additive and so a boundedly finite Borel measure. Finally, it follows from (ii) of Proposition A2.6.11 that Ilk --+.v Jl. To establish part (ii) of the theorem examine the proof of Proposition A2.5.IV. Let C(J' be the class of sets A for which A is a .16'(A[f[ )-measurable mapping into [0, oo ). Again, C(l' is a monotone class containing all bounded open and closed sets on fll', and hence .16'[1[ as well as any ring or semiring generating .16'[1[. Also, if!/' is a semiring of bounded sets generating .16'[1[, and A is .16'll;l-measurable for A E !/' and some a-ring f7l of sets on .1l[i[, then A is !7l-measurable for A E .16'[1[. The proposition now implies that .s;~ The final extension is of the compactness criterion of Theorem A2.4.1. Proposition A2.6.IV. A family of measures {Jl,c} in .Jl[f[ is relatively compact in thew-topology on .Jl[f[ if and only if their restrictions {Jl~n>} to a sequence of closed spheres Sn j fll' are relatively compact in the weak topology on As , in which case the restrictions {Jl:} to any closed bounded Fare relatively compact in the weak topology on .AF. PROOF. Suppose first that {Jla} is relatively compact in thew-topology on .il[f[ and that F is a closed bounded subset of fl£. Given any sequence of the Jl:, there exists by assumption a w-convergent subsequence, Jlak --+.v Jl say. From Proposition A2.6.11, arguing as in the proof of A2.3.11, it follows that for all bounded closed sets C, lim supk--+oo Jlak(C):::;; Jl(C). Hence, in particular, the values of Jlak(F) are bounded above. Moreover, the restrictions {Jl:J are uniformly tight, this property being inherited from their uniform tightness on a closed bounded sphere containing F. Therefore, the restrictions are relatively compact as measures on F, and there exists a further subsequence converging weakly on F to some limit measure, {l say, on F. This is enough to show that the Jl: themselves are relatively compact. Conversely, suppose that there exists a family of spheres Sn, closed or otherwise, such that {Jl~n>} are relatively compact for each n. By diagonal selection, we may choose a subsequence r:xk such that Jl~~ --+ Jl outside a bounded set, then Jf d11~~ --+ Jf d!l The criterion for weak compactness on each Sn can be spelled out in detail from Prohorov's theorem A2.4.1. A particularly neat result obtains in the case that f!l' is locally (and hence countably) compact, when the following terminology is standard. A Radon measure in a locally compact space is a measure taking finite values on compact sets. A sequence {Ilk} of such mea• sures converges vaguely to 11 if Jf dllk --+ Jf d11 for each continuous f vanishing outside a compact set. Now any locally compact space with a countable base is metrizable, but the space is not necessarily complete in the metric so obtained. If, however, the space is both locally compact and a c.s.m.s., it can be represented as the union of a sequence of compact sets Kn with Kn s K~+l, and then by changing to an equivalent metric if necessary, we can ensure that the spheres Sn are compact as well as closed [e.g., see Proposition 2.61 of Hocking and Young (1961)]; we assume this is so. Then a Borel measure is a Radon measure if and only if it is boundedly finite, and vague convergence coincides with *-convergence. The discussion around (A2.6.1) shows that the vague topology is metrizable and suggests one form for a suitable metric. Finally, Proposition A2.6.1V takes the following form. Corollary A2.6.V. If f!l' is a locally compact c.s.m.s., then the family {lla} of Radon measures on fllff is relatively compact in the vague topology if and only if the values {lla(A)} are bounded for each bounded Borel set A. PROOF. Assume the metric is so chosen that closed bounded sets are compact. Then if the J.t,.( ·) are relatively compact on each Sn, it follows from condition (i) of Theorem A2.4.1 that the lla(Sn) are bounded and hence that the IliA) are bounded for any bounded Borel set A. Conversely, suppose the boundedness condition holds. Then, in particular, it holds for Sn, which is compact so the tightness condition (ii) of Theorem A2.4.1 is satisfied trivially. Thus, the {lla} are relatively compact on each Sn and so by Proposition A2.6.IV are relatively compact in the *- (i.e., vague) topology. D A2.7. Measures on Topological Groups A group f§ is a set on which is defined a binary relation f§ x f§ --+ f§ with the following properties: (i) For all g1 , g2 , g3 E f'§, (g1g2 )g 3 = g1 (g2 g3 ) (associative law); (ii) There exists an identity element e (necessarily unique) such that for for all g E f'§, ge = eg = g; 632 APPENDIX 2. Measures on Metric Spaces (iii) For every g E ~there exists a unique inverse g-1 such that g-1 g = gg-1 = e. The group is Abelian if it also has the following property: (iv) For all g1, g2 E ~. g1g2 = g2g1 (commutative law). A homomorphism between groups is a mapping ff that preserves the group operations, in the sense that (ffgd(ffg2) = ff(g1g2) and (ffg}-1 = ffg-1. If the mapping is also one-to-one it is an isomorphism. An automorphism is an isomorphism of the group onto itself. A subgroup Yt' of ~ is a subset of ~ that is closed under the group operations and so forms a group in its own right. If Yt' is nontrivial (i.e., neither {e} nor the whole of~). its action on~ splits~ into equivalence classes, where g1 =g 2 if there exists hE Yt' such that g2 = g1h. These classes form the left cosets of~ relative to Yt'; they may also be described as the (left) quotient space ~/Yt' of~ with respect to Yt'. Similarly, Yt' splits~ into right cosets, which in general will not be the same as the left cosets. If~ is Abelian, however, or more generally if Yt' is a normal (or invariant) subgroup, which means that for every g E ~. h E .Yt', g-1 hg E .Yt', then the right and left cosets coincide, and moreover the products of two elements, one from each of any two given co sets, fall into a uniquely defined third coset. With this definition of multiplication, the cosets then form a group in their own right, namely, the quotient group. The natural map taking an element from ~ into the coset to which it belongs is then a homomorphism of~ into ~/Yt', of which Yt' is the kernel, that is, the inverse image ofthe identity in the image space ~/Yt'. The direct product of two groups ~ and .Yt', written ~ x Yt', consists of the Cartesian products of<'§ and .Yl' with the group operation (g1, h1)(g2, h2) = (g1g2, h1h2), identity (e~, eJt'), and inverse (g, h)-1 = (g-1, h-1 ). In particular, if~ is a group and Yt' a normal subgroup, then ~ is isomorphic to the direct product Y{' X ~j.Y{'. ~ is a topological group if it has a topology il/1 with respect to which the mapping (g 1 , g2 )--+ g 1 g21 from~ x ~(with the product topology) into ~is continuous. This condition makes the operations of left (and right) multi• plication by a fixed element of~. and of inversion, continuous. A theory with wide applications results if the topology lflt is taken to be locally compact and second countable. It is then metrizable, but not necessarily complete in the resulting metric. In keeping with our previous discussion, however, we fre• quently assume that~ is a complete separable metric group (c.s.m.g.), as well as being locally compact. If, as may always be done by a change of metric, the closed bounded sets of~ are compact, we refer to ~ as a u-group. Definition A2.7.1. A u-group is a locally compact, complete separable metric group, with the metric so chosen that closed bounded sets are compact. In this context boundedly finite measures are Radon measures and the concepts of weak and vague convergence coincide. A boundedly finite measure A2.7. Measures on Topological Groups 633 f.L on the u-group is left-invariant if (writing gA = {gx: x E A}) (A2.7.1) f.L(gA) = f.L(A) (g E '§,A E :14'§), or equivalently, (A2.7.2) f"§f(g- 1 x)f.L(dx) = f"§f(x)f.L(dx) for all f E BC('§), the class of continuous functions vanishing outside a bounded (in this case, compact) set. Right-invariance is defined similarly. A fundamental theorem for locally compact groups asserts that up to scale factors they admit unique left- and right-invariant measures, called Haar measures. If the group is Abelian, the left and right Haar measures coincide, as they do also when the group is compact, in which case the Haar measure is totally finite and is uniquely specified when normalized so as to have total mass unity. On the real line, or more generally on !Rd, the Haar measure is just Lebesgue measure t( · ), and the uniqueness referred to above is effectively a restatement of results on the Cauchy functional equation (see also Exercise 10.1.8). If '§ is a topological group and Yl' a subgroup, the quotient topology on t§j.Yl' is the largest topology on t§j.Yl' making the natural map from'§ into t§j.Yl' continuous. It is then also an open map (i.e., takes open sets into open sets). If it is closed, then the quotient topology for '§I Yl' inherits properties from the topology for '§: it is Hausdorff, or compact, or locally compact, if and only if'§ has the same property. These concepts extend to the more general context where fl£ is a c.s.m.s. and Yl' defines a group of one-to-one bounded continuous maps T, of fl£ onto itself, such that T,, (T,,{x)) = T,,h,(x). Again we assume that Yl' is au-group and that the {T,} act continuously on fl£, meaning that the mapping (h, x)-+ T,.(x) is continuous from Yl' x fl£ into fl£. The action of Yl' splits fl£ into equivalence classes, where x 1 =x 2 if there exists hE Yl' such that x 2 = T,(xd. It acts transitively on fl£ if for every x 1 , x 2 E fl£ there exists an h such that T,. maps x 1 into x 2 • In this case the equivalence relation is trivial: there exists only one equivalence class, the whole space fl£. In general, the equivalence classes define a quotient space !2, which may be given the quotient topology; with this topology the natural map taking x into the equivalence class containing it is again both continuous and open. If the original topology on Yl' is not adjusted to the group action, however, the quotient topology may not be adequate for a detailed discussion of invariant measures. EXAMPLE A2.7(a). Consider IR 1 under the action of scale changes: x-+ lXX (0 < IX < oo ). Here Yl' may be identified with the positive half-line (0, oo) with multiplication as the group action. There are three equivalence classes, 634 APPENDIX 2. Measures on Metric Spaces ( -oo, 0), {0}, and (0, oo), which we may identify with the three-point space !2 = { -1, 0, 1}. The quotient topology is trivial (only 0 and the whole of !2), whereas the natural topology for further discussion is the discrete topology on !2, making each of the three points both open and closed in !2. With this topology the natural map is open but not continuous. It does have, however, a continuous (albeit trivial) restriction to each of the three equivalence classes and therefore defines a Borel mapping of !!l" into !2. An important problem is to determine the structure of boundedly finite measures on !!l", which are invariant under the group of mappings {1/,}. In many cases, some or all of the equivalence classes of !!l" under :Ye can be identified with replicas of :Ye, so that we may expect the restriction of the invariant measure to such cosets to be proportional to Haar measure. When such an identification is possible, the following simple lemma can be used; it allows us to deal with most of the situations arising from concrete examples of invariant measures [for further background see, e.g., Bourbaki (1963)]. Lemma A2.7.11. Let!!{= :Ye x !lJI, where :Ye is au-group and !lJI is a c.s.m.s., and suppose that J.l e Jlfll" is invariant under left multiplication by elements of :Ye, in the sense that for A E f!lfl£, B e f!l!!lf, (A2.7.3) f.l(hA X B) = J.l(A X B). Then J.l = t x K, where tis a multiple of left Haar measure on :Ye and K e Jl'!l/ is essentially uniquely determined. PROOF. Consider the set function f.lB( ·)defined on f!IJY for fixed Be f!l!!lf by J.lB(A) = J.l(A X B). Then J.lB inherits from J.l the properties of countable additivity and bounded finiteness, and so defines an element of JIJY. But then, from (A2.7.3), J.lB(hA) = f.l(hA X B) = J.l(A X B) = J.lB(A), implying that J.lB is invariant under left multiplication by elements of :Ye. It therefore reduces to a multiple of left Haar measure on :Ye, say J.lB(A) = K(B)t(A}. Now the family of constants K(B) may be regarded as a set function on f!l!!lf, and, as for J.lB, this function is both countably additive and boundedly finite. Consequently, K( ·) e vii !!If, and it follows that J.l(A X B) = J.lB(A) = t(A)K(B). In other words, J.l reduces to the required product form on product sets, and since these generate f!lfl£, f.l and the product measure t x K coincide. D To apply this result to specific examples it is often necessary to find a suitable product representation for the space on which the transformations act. The situation is formalized in the following statement. A2.7. Measures on Topological Groups 635 Proposition A2.7.111. Let !'£ be a c.s.m.s. acted on measurably by a group of transformations {1;,: hE .Yt'}, where .Yt' is a a-group. Suppose, furthermore, that there exists a mapping ljJ: .Yt' x OJ/ --+ !'£,where OJ/ is a c.s.m.s. and ljJ is one-to-one, both ways measurable, takes bounded sets into bounded sets, and preserves the transformations {1;,} in the sense that (A2.7.4) ~-!/J(h, y) = ljJ(h' h, y). Let J1 be a measure on .II!!f which is invariant under the transformation ~· Then there exists a unique measure K E .II"!! such that, for &6!!f-measurable nonnegative functions f, (A2.7.5) tf(x)Jl(dx) = t K(dy) fxf(ljJ(h, y))t(dh). PROOF. Let fi be the image of J1 induced on .Yt' x OJ/ by the mapping l/J; that is, ji(A x B) = Jl(!/J(A x B)). Then fi(hA X B) = Jl(!/J(hA X B)) = Jl(~!/J(A X B)) = Jl(!/J(A X B)) = fi(A X B), so that fi is invariant under the action of h E .Yt' on the first argument. Moreover, if A and B are bounded sets in .Yt' and OJ/, respectively, then by assumption !/J(A x B) is bounded in!'£, so that fi is boundedly finite whenever J1 is boundedly finite. Lemma A2.7.11 can now be applied and yields the result that ji(A X B) = t(A)K(B) for some unique boundedly finite measure Kin .II"!!. This relation establishes the truth of (A2.7.5) for indicator functions Il/I EXAMPLE A2.7(b). Let J1 be a measure on IR 2 , which is invariant under rotations about the origin. These may be written 76 for() E §, § being the circumference of the unit disk with addition modulo 2n. The equivalence classes consist of circles of varying radii centred on the origin, together with the isolated point {0}. The mapping (r, ())--+ (r cos(), r sin()) takes the product space § x IR+ into IR 2 \ {0} and is a representation of the required kind for IR 2 \ {0}. We therefore write J1 as the sum of a point mass at the origin and a measure on IR2 \ {0}, which is invariant under rotations and can therefore be represented as the image of the uniform distribution around the circle and a measure K on the positive half-line. Integration with respect to J1 takes the form [cf. (A2.7.5)] 2 r f(X)f.l(dx) = f(O)f.l{O} + Joo K(dr) f " f(r cos(), r sin ())(2n)-1 d(). J[R2 0+ 0 636 APPENDIX 2. Measures on Metric Spaces A2.8. Fourier Transforms In this section we collect together a few basic facts from classical Fourier transform theory. For brevity, most results are stated for Fourier transforms of functions on IR =IR 1; the corresponding results for !Rd can be obtained by no more than changes in the domain of integration and appropriate book• keeping with multiples of 2n. Both the !Rd theory and the theory of Fourier series, which can be regarded as Fourier transforms of functions defined on the unit circle, are subsumed under the concluding comments concerned with Fourier transforms of functions defined on locally compact Abelian groups. We refer to texts such as Titchmarsh (1937) for more specific material on these topics. For any real- or complex-valued measurable (Lebesgue) integrable function f( ·),its Fourier transform j( ·)is defined by (A2.8.1) j(w) = J: eiwxf(x)dx (wE IR). Iff is real and symmetric then so is j In any case,j is bounded and continu• ous, while the Riemann-Lebesgue lemma asserts that f(w)-+ 0 as lwl-+ oo. Furthermore, if j is integrable then the inverse relation (A2.8.2) f(w) = (2nr1 J: eixwj(w)dw holds. The theory is not symmetric with respect to f and j: for example, see Titchmarsh (1937) for a more detailed account of the representation of a function by its inverse Fourier transform. A symmetrical theory results if we consider (real- or complex-valued) functions that are square integrable. We have the Plancherel identities for square integrable functions f and g: (A2.8.3) J: f(x)g(x)dx = 2~ J: j(w)g(w)dw, and, with g = f, ao 1 fao (A2.8.4) f -co lf(xWdx = 2n -ao IRwWdw. Here the Fourier transform cannot be obtained directly from (A2.8.1) but can be represented as a mean square limit (A2.8.5) j(w) = l.i.m. fT eiwxf(x)dx, T-+ao -T the existence of the finite integral following readily from the Schwarz inequal• ity. Since the limit is defined only up to an equivalence, the theory is strictly A2.8. Fourier Transforms 637 between equivalence classes of functions, that is, elements of the Hilbert space L2 (1R), rather than a theory between individual functions. An important version for probability theory is concerned with the Fourier transforms of totally finite measures (or signed measures). If G is such a measure, its F ourier-Stieltjes transform g is the bounded uniformly con• tinuous function (A2.8.6) g(w) = J: eiwxG(dx). If G is a probability measure, g(w) is its characteristic function and g is then a positive-definite function: for arbitrary finite families of real numbers w 1, ••• , w, and complex n urn hers IX 1 , ••• , IX., r r (A2.8.7) L L IX;aig(w; - wi) ~ 0. i=1 j=1 Conversely, Bochner's theorem asserts that any function continuous at w = 0 and satisfying (A2.8.7) can be represented as the Fourier transform of a totally finite measure G on IR, with G(IR) = g(O). If we take any real or complex integrable function f with any totally finite signed measure G and apply Fubini's theorem to the double integral t: t: eiwxf(w)G(dx)dw, which is certainly well defined, we obtain Parseval's identity (A2.8.8) J: j(x)G(dx) = J: f(w)g(w)dw. This identity is of basic importance, since it shows that G is uniquely deter• mined by g. Various more specific inversion theorems can be obtained by taking suitable choices off followed by a passage to the limit: this approach is outlined in Chapter XV, Section 3, of Feller (1966), for example. In partic• ular, the following two forms are traditional: (i) for continuity intervals (a, b) of G, G((a, b)) = lim IT (iwt1(e-iwa- e-iwb)g(w)dw; T-+oo -T (ii) for an atom a of G, G({a}) = lim (2Tt1fT e-iwag(w)dw. T-+oo -T Much of the preceding theory can be extended without difficulty from IR to the case of a locally compact Abelian topological group '§. The characters of such a group are the continuous homomorphisms of the group onto the complex numbers of modulus 1. If x1 , x2 are characters, then so are x1 x2 and 638 APPENDIX 2. Measures on Metric Spaces x1 1 . Thus, the characters form a group in their own right, {9 say, the dual group for~- There is a natural topology on {9, namely, the smallest making the evaluation mapping e9 (X) =x(g) continuous for each g E ~. and with this topology {9 also is a locally compact Abelian topological group. If~ = ~. the characters are of the form e1"'x (w E ~), and {9 can be identified with another version of ~- If~ = 7!.., the group of integers, {9 is the circle group, and vice versa. In any case, the original group reappears as the dual of the dual group {9, and if~ is compact, {9 is discrete and conversely. Now let H and 11 denote Haar measure on ~ and {9, respectively. If f: ~--+ ~ is measurable and H-integrable, its Fourier transform j is the function defined on {9 by (A2.8.9) f(x) = 1, x(g)f(g)H(dg). If also j is 11-integrable, then the inverse relation (A2.8.10) f(x) = J~ x(g)f(x)l1(dx) holds, provided that 11 is normed appropriately [otherwise, a normalizing constant such as (2nf1 in (A2.8.2) is needed]. Assuming that such a norming has been adopted, the appropriate analogues of(A2.8.4)-(A2.8.8) remain true. In particular, we note the generalized Plancherel identity (A2.8.11) f~ lf(g)l2 H(dg) = f~ i](xW 11(dx). APPENDIX 3 Conditional Expectations, Stopping Times, and Martingales This appendix contains mainly background material for Chapter 13. For further discussion and most proofs we refer the reader to Ash (1972), Chung (1974), and Bremaud (1981) and to various references cited in the text. A3.1. Conditional Expectations Let (Q, Iff, .9') be a probability space (see Section A1.4), X a random variable (r.v.) with EIXI = fn IXIY'(dw) < oo, and I§ a sub-u-algebra of events from iff. The conditional expectation of X with respect to !:§, written E(XI'§) or else Ex 1\'#(w), is the '§-measurable function (i.e., a random variable) defined up to values on a set of!:§ of .9'-measure zero as the Radon~ Nikodym derivative E(XI'§) = Ex 1\'#(w) = ~!f>(dw)j&<\'#>(dw), where ~x(A) =fA X(w).9'(dw) is the indefinite integral of X and the superscript ('§) indicates that the set functions are to be restricted to !:§. It is clear that ~if> « ,9'<\9'> so that the Radon~ Nikodym derivative exists as a '§-measurable function. Since, in general,!:§ is by assumption a coarser o--algebra than Iff, the r.v. Ex 1\'#(w) represents a smoothed version of the original r.v. X(w). The '§-measurability of E(XI'§) implies that (A3.1.1) X(w).9'(dw) Ex E t = t 1\'#(w)&(dw) (all U '§), an equation that determines the conditional expectation uniquely and is usually taken as the defining relation. Extending (A3.1.1) from '§-measurable indicator functions lu(w) to more general '§-measurable functions Y, we have, 640 APPENDIX 3. Conditional Expectations, Stopping Times, and Martingales whenever EIXI and EIXYI exist, (A3.1.2) E(XY) = t Y(w)X(w)&(dw) = t Y(w)Ex 1 ~(w)&(dw) = E(YE(XIC#)). Now replacing Y by lu Y (U E C#) and using (A3.1.1), there follows the factori• zation property of conditional expectations, that for C#-measurable r.v.s Y, for which both EIXI and EIXYI exist, (A3.1.3) E(XYIC#) = YE(XIC#) a.s. Conditional expectations inherit many standard properties of ordinary expectations: (A3.1.4) linearity: ECt cxiXile#) =it cxiE(Xile#); (A3.1.5) monotonicity: X ::s;; Y a.s. implies E(XIC#) ::s;; E(YIC#) a.s.; (A3.1.6) monotone convergence: xn ~ 0, xn i y a.s. implies E(Xnle#) i E(YIC#) a.s.; (A3.1.7) Jensen's inequality: For convex measurable functions f: IR--+ IR with Elf(X)I < oo, f(E(XIC#)) ::s;; E(f(X)IC#) a.s. [convexity here means that f((x + y)/2) ::s;; (f(x) + f(y))/2]. If ~1 and ~2 are two sub-a-algebras with ~1 <:;: ~2 <:;: .9' and EIXI < oo as before, the repeated conditioning theorem holds: (A3.1.8) yielding as the special case when C#1 = {0, 0}, (A3.1.9) E[E(XIC#)] = EX. Two a-algebras <;'§ and £' are independent if, for all A E C# and BE £', &(A n B) = .'?Jl(A).'?Jl(B). Given such <;'§ and £', if X is <;'§-measurable and we seek E(XI£'), we may expect it to reduce to yield (A3.1.10) E(XI£') =EX. This is a special case of the principle of redundant conditioning: if the r.v. X is independent of£' [i.e., a(X) and £' are independent a-algebras] and C# is independent of£', then (A3.1.11) E(XIC# v £') = E(XIC#), reducing to (A3.1.10) for trivial C#. Let fl be a c.s.m.s. and X an f£-valued r.v. on (0, tff, .9). Given a sub-a• algebra <;'§ of tff, the conditional distribution of X given <;'§is defined by analogy with (A3.1.1) by (A3.1.12) &(X E Ale#)= E(JA(X)IC#) (A E ~(X)). A3.1. Conditional Expectations 641 As in Section Al.5, the question of the existence of regular conditional distri• butions arises. In our present context, we seek a kernel function Q(A, w) (A E 96'(~}, W E 0) such that for fixed A, Q(A, ·)is a ~-measurable function of w [and we identify this with (A3.1.12}], while for fixed w, we want Q( ·, w) to be a probability measure on 96'(~). Introduce the set function n( ·)defined initially for product sets A x U for A E 96'(~). U E ~. by (A3.1.13) n(A x U) = LIA(X(w)).?J'(dw). Since n( ·) is countably additive on such sets, it can be extended to a measure, clearly a probability, on (~ x 0, 96'(~) ® ~). Then Proposition Al.5.III can be applied and yields the following formal statement, in which we identify the kernel function Q( ·, ·)sought above with .?JI(X E AI~). Proposition A3.1.1. Let ~ be a c.s.m.s., (0, If, .?1') a probability space, and X an ~-valued r.v. defined on (0, If, .?1'). If~ is a sub-a-algebra of If, then there exists a regular version of the conditional distribution &'xe·l~(w) such that (i) .?l'xe·l~(w) is a probability measure on 91(~) for each fixed w; (ii) .?l'xeAI~( ·)is a ~-measurable function of w for fixed A E 96'(~); (iii) for each U E ~ and A E 96'(~). (A3.1.14) L&'xeAi~(w).?J'(dw) = LIA(X(w)).?J'(dw). Observe that if~= If, then the conditional distribution .?l'xe·l~(w) is the degenerate distribution concentrated on the point X(w). In general, the con• ditional distribution represents a blurred image of this degenerate distribu• tion, the blurring arising as a result of the incomplete information concerning X carried by the sub-a-algebra ~. Consider the following question, which is of the nature of a converse to the proposition. Given(~. 96'(~)), (0, If, .?1'), and a regular kernel Q(A, w), can we find a refinement If' 2 If and an $'-measurable ~-valued r.v. X such that Q(A, w) coincides with .?l'xeAI~(w)? If we confine ourselves to the original space this may not necessarily be possible, but by extending 0 we can accomplish our aim. Take the probability space (0', If', .?1'') given by 0' = ~ x 0, If' = 96'(~) ®If, .?1'' = n as constructed via (A3.1.13) (identifying ~ there with If here), and consider the r.v. X:~ x 0-+ ~for which X(w') = X(x, w) = x. With the mapping T: 0'-+ 0 for which T(w') = T(x, w) = w, so that T-1 (1!) is a sub-a-algebra of If', we then have, for A E 96'(~). (A3.1.15) .?JI~eAIT-'(Cl(w') = Q(A, T(w')) = Q(A, w). Often the conditioning a-algebra~ is itself generated by some real or (more generally) c.s.m.s.-valued r.v. Y. Then E(XI~) is called the conditional expec- 642 APPENDIX 3. Conditional Expectations, Stopping Times, and Martingales tation of X given Y, and &I(X E AI~) the conditional distribution of X given Y, together with the suggestive notation E(XI Y) or Ex1r(w) and &I(X E AI~) or &lxeAI~(w). Equation (A3.1.3) then implies, for any Borel-measurable func• tion h( ·) such that the unconditional expectations exist, (A3.1.16) E(Xh(Y)I Y) = h(Y)E(XI Y). The terminology suggests that, although E(XI Y) is defined as a r.v., its value should depend on w only through Y(w). Thus, if Y takes its values in a c.s.m.s. 11JJ, we should look for a real-valued .11(11JJ)-measurable function hx1r(Y) such that (A3.1.17) Exlr(w) = hxlr(Y(w)) a.s. That such a function exists is the assertion of the Doob representation theorem (e.g., see Doob, 1953). It can be established by applying the argument around (A3.1.1) to the measures induced on .11(11JJ) by the equations (B E .11(11JJ)}, ~x(B) = f X(w)&l(dw), y-I(B) and, noting that ~x « &lr on .11(11JJ), by applying the Radon-Nikodym theorem. Since the product of a finite or denumerably infinite number of c.s.m.s.s can itself be regarded as a c.s.m.s., we state the theorem in the following general form. Proposition A3.1.11. Let (Q, Iff, &I) be a probability space, X an integrable real-valued r.v. on n, and~ a sub-u-algebra of 8 generated by a countable family of r.v.s. Y = {Y 1 , Y2 , ••• } taking their values in the c.s.m.s.s 11JJ1 , 11JJ2 , ••• , respec• tively. Then there exists a Borel measurable function hxlr( · ): 11JJ1 x 11JJ2 x · · · --+ ~.such that (A3.1.18) The proposition concerning regular conditional distributions can be trans• formed in a similar way, yielding a kernel &lxeAIY(y 1, y2 , •• • ), which is a probability distribution in A for each vector (y1, y2 , •• • ), a Borel-measurable function of the family (y 1 , y2 , ••• ) for each A, and satisfies &lxeAI~(w) = &lxeAir(Yl(w}, Yz(W), ... ) &1-a.s. When densities exist with respect to some underlying measure J1. such as Lebesgue measure on ~d. the conditional distributions have the form /1}j ( ) _ JAfx.r(x, Y1• Yz, ... )JJ.(dx) A3.2. Convergence Concepts Most of the different notions of convergence and of uniform integrability mentioned below are standard. Stable convergence is less familiar and is discussed in more detail. A sequence of r.v.s {Xn: n = 1, 2, ... } on a common probability space (Q, c!, &')converges in probability to a limit r.v. X, also defined on (Q, c!, &'),if for all e > 0, (A3.2.1) &'{IXn- XI > e}--+ 0 as n--+ oo. The sequence converges almost surely to X if (A3.2.2) 1 = &'{w: Xn(w)--+ X(w) (n--+ oo)} = &'Co nQ mQn {w: IXm(w) - X(w)l < r-1}) =&'Co n01 mQn {w: IXm(w)- Xn(w)l < r-1})· Both these concepts readily generalize to the case where the r.v.s X and Xn are q"-valued for some c.s.m.s. q- by simply replacing the Euclidean distance IX - Yl by the metric p(X, Y) for X, Y E q-. The a.s. convergence as at (A3.2.2) implies convergence in probability; convergence in probability implies the existence of a subsequence {XnJ that converges a.s. to the same limit. Returning to the real-valued case, for any given real p;;::: 1, {Xn} converges in the mean of order p (or in pth mean, or in LP norm), if the pth moments exist and (A3.2.3) (n--+ oo ), the norm here denoting the norm in the Banach space Lp(Q, c!, &') of equiv• alence classes of r.v.s with finite pth moments. For p = oo, the space L 00 (Q, c!, &')consists of &'-essentially bounded r.v.s X, that is, r.v.s X for which lXI ::s;; M a.s. for some M < oo; then (A3.2.4) IIXIIoo = esssupiX(w)l = inf{M: IX(w)l ::s;; M a.s.}. If Xn --+X in pth mean, then E(X:) --+ E(XP) (n --+ oo ). Chebyshev's inequality, in the form for an LP r.v. X, (A3.2.5) &'{IX- al > e} ::s;; e-PE(IX- alP) (real a, e > 0), implies that convergence in LP norm implies convergence in probability. The converse requires the additional condition of uniform integrability. Definition A3.2.1. A family of real-valued r.v.s {X,: t E ff} defined on the common probability space (Q, c!, &') is uniformly integrable if, given e > 0, there exists M < oo such that (A3.2.6) r IX,(w)l&'(dw) < e (all t E ff). JIX,I>M 644 APPENDIX 3. Conditional Expectations, Stopping Times, and Martingales Proposition A3.2.11. Let the r.v.s {Xn: n = 1, 2, ... } and X be defined on a common probability space (Q, tff, &') and be such that Xn -+ X in probability. Then a necessary and sufficient condition for the means to exist and for Xn -+ X in L1 norm is that the sequence {Xn} be uniformly integrable. Applied to the sequence {X:} and noting the inequality EIXn- XIP::::;; 2P(EIXniP + EIXIP) (1 ::::;; p < oo), the proposition extends in an obvious way to convergence in LP norm for 1 ::::;; p < oo. A weaker concept than convergence in LP norm [i.e., strong convergence in the Banach space Lp(Q, tff, &')] is that of weak LP convergence; namely, that if Xn, X E LP, then E(Xn Y)-+ E(XY) (n-+ oo) for all Y E Lq, where p-l + q-l = 1. When Xn is El-valued for a c.s.m.s. El with metric p, Xn converges to X in distribution if &{Xn E A}-+ &{X E A} for all A E aJ(El) for which &{X E 8A} = 0. This type of convergence is not a constraint on the r.v.s so much as a constraint on the distributions they induce on aJ(El): indeed, it is precisely the weak convergence of their induced distributions. If Xn -+X in probability (or, a fortiori, if Xn-+ X a.s. or in LP norm), then from the inequalities &I'{Xn E A}- &'{X E A} ::::;; &I'({Xn E A} 11 {X E Ac}) ::::;; &({Xn E A} 11 {X E (A')<})+ &{X E A'}- &{X E A} ::::;; &l'{p(Xn, X)> e} +&{X E A'}- &{X E A}, it follows that Xn-+ X in distribution, also written Xn -+d X. No general converse statement is possible except when X is degenerate, that is, X = a a.s. for some a EEl. For this exceptional case, Xn -+d a means that &{p(Xn, a)< e} = &I' { Xn E S(a; e)} -+ 1 (n -+ oo ), where S(a; e) is the sphere with centre a and radius e, which is the same as Xn-+ a in probability. A hybrid concept, in the sense that it depends partly on the r.v.s Xn themselves and partly on their distributions, is that of stable convergence. Definition A3.2.111. If {Xn: n = 1, 2, ... } and X are El-valued r.v.s on (Q, tff, &') and fF is a sub-a-algebra of tff, then Xn -+X (iF -stably) in distribution iffor all U E fF and all A E aJ(El) with &{X E oA} = 0, (A3.2.7) &'( {Xn E A} 11 U)-+ &'({X E A} 11 U) (n-+ oo). The hybrid nature of stable convergence is well illustrated by the facts that when fF = {0, n}, iF-stable convergence is convergence in distribution whereas when fF 2 u(X), we have a.s. convergence in probability, because the regular version &'xe·l9'(w) of the conditional distribution appearing in &({X E A} 11 U) = Ju&xeAI9'(w)&(dw) can be taken as being {0, 1}-valued, and when such degenerate distributions for the limit r.v. occur the concepts of convergence in distribution and in probability coincide as already noted. A3.2. Convergence Concepts 645 In general, stable convergence always implies weak convergence, and it may be regarded as a form of weak convergence of the conditional distributions &'(Xn E A Iff). Just as weak convergence can be expressed in equivalent ways, so also can stable convergence, as set out below (see Aldous and Eagleson, 1978). Proposition A3.2.IV. Let {Xn}, X, ff be as in Definition A3.2.111. Then the following conditions are equivalent: (i) Xn-+ X ($'-stably) [i.e., (A3.2.7) holds). (ii) For all $'-measurable &'-essentially bounded r.v.s Z and all bounded con• tinuous h: fiE-+ IR, (A3.2.8) E(Zh(Xn)) -+ E(Zh(X)) (n-+ oo). (iii) For all real-valued ff -measurable r.v.s Y, the pair (Xn, Y) converges jointly in distribution to the pair (X, Y). (iv) For all bounded continuous functions g: fiE x IR-+ IR and all real-valued ff -measurable r.v.s Y, (A3.2.9) g(Xn, Y) -+ g(X, Y) (31' -stably). If fiE= !Rd then any of (i)-(iv) is equivalent to condition (v): (v) For all real vectors t E !Rd and all &'-essentially bounded ff -measurable r.v.s Z, (A3.2.10) E(Z exp(it' Xn)) -+ E(Z exp(it' X)). PROOF. Equation (A3.2.7) is the special case of (A3.2.8) with Z = Iu(w) and h(x) = IA(x) for U E ff and A E fJI(fi£), except that such h( ·)is not in general continuous: as in the continuity theorem for weak convergence, (A3.2.8) can be extended to the case where h is bounded and Borel measurable and &'{X E oh} = 0, where oh is the set of discontinuities of h. When fiE= !Rd, (A3.2.10) extends the well-known result that joint convergence of charac• teristic functions is equivalent to weak convergence of distributions. Note that all of (A3.2.7), (A3.2.8), and (A3.2.10) are contracted versions of the full statement of weak convergence in L 1 of the conditional distributions, namely, that (A3.2.11) E(ZE(h(Xn)lff))-+ E(ZE(h(X)Iff)) (n-+oo) for arbitrary (not necessarily ff -measurable) r.v.s Z. However, (A3.2.11) can immediately be reduced to the simpler contracted forms by using the repeated conditioning theorem, which shows, first, that it is enough to consider Z ff -measurable, and second, that when Z is ff -measurable the conditioning on ff can be dropped. If Y is real-valued and $'-measurable and in (A3.2.7) we set U = y-1(B) for B E fJI(IR), we obtain &'{(Xn, Y)EA X B}-+&'{(X, Y)EA X B}, from which (iii) follows. Conversely, taking Y = Iu in (iii) yields (A3.2.7). 646 APPENDIX 3. Conditional Expectations, Stopping Times, and Martingales Finally, for any two real-valued ff-measurable r.v.s Y, Z, repeated appli• cation of (iii) shows that (X,., Y, Z) converges weakly in distribution to the triple (X, Y, Z). Applying the continuous mapping theorem (Proposition A2.2.VII) yields the result that the pairs (g(X,., Y), Z) converge weakly in distribution to (g(X, Y), Z), which is equivalent to the stable convergence of g(X,., Y) to g(X, Y) by (iii). Since stable convergence implies weak conver• gence, (iv) implies (iii). D When the limit r.v. is independent of the conditioning a-algebra ff, we have a special case of some importance. Then, (A3.2.7) and (A3.2.10) reduce to the forms (A3.2.12) ?J!(X,. E AI U)-+ ?J!{X E A} (?J!(U) > 0) and (A3.2.13) E(Z exp(it' X,.))-+ (EZ)E(exp(it' X)), respectively. In this case the X,. are said to converge ff-mixing to X. In applications it is often the case that the left-hand sides of relations like (A3.2.7) converge as n-+ oo, but it is not immediately clear that the limit can be associated with the conditional distribution of a well-defined r. v. X. Indeed, in general there is no guarantee that such a limit r.v. will exist, but we can instead extend the probability space in such a way that on the extended space a new sequence of r.v.s can be defined, with effectively the same conditional distributions as for the original r.v.s, and for which there is ~-stable con• vergence in the limit to a proper conditional distribution. Lemma A3.2.V. Suppose that for each U E ~ and for A in some covering ring generating .?4(2l), the sequences {?J!( {X,. E A} n U)} converge. Then there exists a probability space (Q', 8', &'), a measurable mapping T: (Q', C')-+ (Q, 8), and a r.v. X' defined on (Q', 8') such that if ff' = r-1 ff and X~(w') = X,.(Tw'), then X~ -+ X' (~'-stably). PROOF. Set Q' = f£ x Q and let 8' be the smallest a-algebra of subsets of Q' containing both .?4(2l) ® ~ and also 2l x 8. Defining T by T(x, w) = w, we see that T is measurable. Also, for each A E .?4(2l) and U E ~. the limit n(A x U) = lim,._, 00 ?J!( {X,. E A} n U) exists by assumption and defines a countably additive set function on such product sets. Similarly, we can set n(2l x B)= lim,._, 00 ?J!( {X,. E 2l} n B)= ?J!(B) for BE 8. Thus, n can be extended to a countably additive set function, ?J!' say, on 8'. Observe that ~' = r-1 ~ consists of all sets 2l x U for U E ~- Define also X'(x, w) = x. Then for U' = 2l x U E ~', ?J!'( {X~ E A} n U') = ?J!( {X,. E A} n U)-+ ?J!'(A x U) = ?J!'({X' E A} n U') so that X~ converges to X', ~'-stably. D A3.2. Convergence Concepts 647 Each of the conditions (i)-(v) of Proposition A3.2.IV consists of a family of sequences, involving r.v.s Xn converging in some sense, and the family of the limits is identified with a family involving a limit r.v. X. It is left to the reader to verify via Lemma A3.2.V that if we are given only the convergence parts of any of these conditions, then the conditions are still equivalent, and it is possible to extend the probability space and construct a new sequence of r.v.s X~ with the same joint probability distributions as the original Xn, together with a limit r.v. X', such that X~~ X', ~'-stably, and so on. In a similar vein, there exists the following selection theorem for stable convergence. Proposition A3.2.VI. Let {Xn} be a sequence of ?£-valued r.v.s on (Q, iff,&'), and ~a sub-(1-algebra of iff. If (i) either~ is countably generated, or~ 2 (J'(X1 , X2 , .•• ); and (ii) the distributions of the {Xn} converge weakly on 81(?£); then there exists an extended probability space (Q', iff',&''), elements T, ~',X~ defined as in Lemma A3.2.V, a sequence {nd, and a limit r.v. X', such that {X~J converges to X', ~'-stably, ask~ oo. PROOF. Suppose first that ~ is countably generated, and denote by [)t some countable ring generating~- For each U E [)t the measures on 81(?£) defined by Qn(A; U) = &'( {Xn E A} n U) are uniformly tight because they are strictly dominated by the uniformly tight measures &'( { Xn E A}). Thus, they contain a weakly convergent subsequence. Using a diagonal selection argument, the subsequence can be so chosen that convergence holds simultaneously for all U E {)t. Therefore, we can assume that the sequence {Qnk(A; U)} converges ask~ oo to some limit Q(A; U) for all A that are continuity sets of this limit measure and for all U E {)t. Given e > 0 and BE~. there exist U., V. E [)t such that U, £ B £ V. and &'(U,) ~ &'(V.)- e. Then the two extreme terms in the chain of inequalities lim Qnk(A; U,) ~lim inf &({Xnk E A} n B) k-+oo k-+oo ~ lim sup &( {Xnk E A} n B) ~ lim QnJA; V.) k-+oo k-+oo differ by at most e, so the sequence {&( { Xnk E A} n B)} also converges. The construction of an extended probability space (Q', iff',&'') and a limit r.v. X' now follows as in the lemma, establishing the proposition in the case that ~ is countably generated. To treat the case that ~ 2 (J'(X1 , X2 , ..• ), consider first the case that ~ = ~0 =(J'(X 1 , X2 , ... ). This is countably generated because ?l' is separable and only a countable family of r.v.s is involved. Applying the selection argu- 648 APPENDIX 3. Conditional Expectations, Stopping Times, and Martingales ment and extension ofthe probability space, we can conclude from (A3.2.10) that (A3.2.14) E(Zh(X~J)--+ E(Zh(X')) for any ~~-measurable Z. Now let Z' be any ~'-measurable r.v. (where ~ ::::l ~0 ). Because h(X~J is ~~-measurable, we can write E(Z'h(X~J) = E(E(Z'I~~)h(X~J), and the convergence follows from (A3.2.14) by the ~~-measurability of E(Z'I~~). Thus, for any such Z', E(Z'h(Z~J)--+ E(Z'h(X')), implying that X~k --+X' (~'-stably). D A systematic account of the topology of stable convergence when ~ = C but no limit r.v. is assumed is given by Jacod and Memin (1984). A3.3. Processes and Stopping Times This section is primarily intended as background material for Chapter 13 where the focus is on certain real-valued stochastic processes denoted {X1(w)} = {X(t, w)} = {X(t)} on the positive time axis, t E (0, oo) =IR+. Other time domains-finite intervals, or IR, or (subsets of) the integers 7L ~ {0, ± 1, ... }-can be considered: it is left to the reader to supply appropriate modifications to the theory as needed. Our aim here is to give just so much of the measure-theoretic framework as we hope will make our text intelligible. For a detailed discussion of this framework, texts like Dellacherie (1972) or Dellacherie and Meyer (1978) or Elliott (1982) should be consulted. Con• densed accounts of selected results, such as given here, are also given in Bremaud (1981), Kallianpur (1980), and Liptser and Shiryayev (1977). While a stochastic process X(t, w) may be regarded as an indexed family of random variables on a common probability space (Q, C, {!IJ), with index set here taken to be IR+, it is more appropriate for our purposes, as in the general theory, to regard it as a function on the product space IR+ x n. The stochastic process X: IR+ x n--+ 14(1R+) ®Cis measurable when this mapping is mea• surable; that is, for all A E 14(1R), (A3.3.1) {(t, w): X(t, w) E A} E 14(1R+) ® C, where the right-hand side denotes the product a-algebra of the two a-algebras there. As a consequence of this measurability and Fubini's theorem, X(·, w): IR+ --+ IR is a.s. measurable, while for measurable functions h: IR --+ IR, Y(w) = l h(X(t, w))dt J~. is a random variable provided the integral exists. A stochastic process on IR+, A3.3. Processes and Stopping Times 649 if defined merely as an indexed family of r.v.s on a common probability space, is necessarily measurable if, for example, the trajectories are either a.s. con• tinuous or a.s. monotonic and right-continuous. The main topic we treat concerns the evolution of a stochastic process; that is, we observe {X(s, w): 0 < s ::5; t} for some (unknown) w and finite time interval (0, t]. It is then natural to consider the cr-algebra $', f}'.(X) ~ $',(X) for 0 < s < t < oo. Of course, we may also have some foreknowledge of the process X, and this we represent by a cr-algebra ~0 . Quite generally, an expanding family ~ = {$',: 0 ::5; t ::5; oo} of sub-cr-algebras of tff is called a filtration or a history, and we concentrate on those histories that incorporate information on the process X. For this purpose, we want the r.v. X(t, w) to be $',-measurable (all t); we then say that X is ~-adapted. We adopt the special notation .;Ye = {$',(X): 0 :$; t :$; 00} := { Jf,: 0 :$; t :$; 00 }, Where ~JXl = liminft>Off,(X) = {0, Q} and ~~X)= nt>O$',(Xl, and Call .;Ye the internal or minimal or natural history of the process X, both of these last two names reflecting the fact that .:Ye is the smallest family of nested cr-algebras to which X is adapted. Any history of the form ~ = {ffo v £,: 0 ::5; t ::5; oo} is called an intrinsic history. Suppose X is measurable and ~-adapted. An apparently stronger condi• tion to impose on X is that of progressive measurability with respect to ~. meaning that for every t E IR+ and any A E ~(IR), (A3.3.2) {(s, w): 0 < s ::5; t, X(s, w) E A} E ~((0, t]) ® $',. Certainly, (A3.3.2) is more restrictive on X than (A3.3.1), and while (A3.3.2) implies (A3.3.1), the converse is not quite true. What can be shown, however, is that given any measurable ~-adapted IR-valued process X, we can find an ~-progressively measurable process Y (that is therefore measurable and ~-adapted) that is a modification of X in the sense of being defined (like X) on (Q, tff, &I) and satisfying (A3.3.3) &l{w: X(t, w) = Y(t, w)} = 1 (all t) [e.g., see Theorems 29 and 30 of Chapter IV ofDellacherie and Meyer (1978)]. The sets of the form [s, t] x U, 0 ::5; s < t, U E $',, t ~ 0, generate a cr• algebra on IR+ X n, which may be called the ~-progressive cr-algebra. Then the requirement that the process X be ~-progressively measurable may be rephrased as the requirement that X(t, w) be measurable with respect to the ~-progressive cr-algebra. A more restrictive condition to impose on X is that it be ~-predictable 650 APPENDIX 3. Conditional Expectations, Stopping Times, and Martingales (the term :F-previsible is also used). Call the sub-a-algebra of BiJ(IR+) ® & generated by product sets of the form (s, t] x U, where U e fi'., t ~ s, and 0::::;; s < oo, the predictable a-algebra, denoted'¥~. (The terminology is well chosen, because it reflects what can be predicted at some "future" time t given the evolution of the process-as revealed by sets U e ff.-up to the "present" time s). Then X is :F-predictable when it is '¥~-measurable; that is, for any A E BiJ(IR), {(t, w): X(t, w) e A} e '¥~. The archetypal :F-predictable process is left-continuous, and this is reflected in Lemma A3.3.1 below, in which the left-continuous history~-~= {3";-} associated with :F appears: here, 31'0 _ = 31'0 and 3";- =lim sups Lemma A3.3.1. An :F-predictable process is ~- 1 -adapted. PROOF. Consider first a process of the form (A3.3.4) X(t, w) = I(a,bl(t)lu(w) (0 < a < b < oo, U E 3";,), which is :F-predictable by construction of'¥~. For given t, {w: X(t, w)) = 1} = 0 if a~ tor b < t, = U if a < t::::;; b, so X(t, w) is 3";_-measurable. Since an arbitrary :F-predictable function can be approximated by a linear combina• tion of functions of this type, and since the class of ~- 1 -adapted processes is closed under linear combinations and monotone limits, standard extension arguments complete the proof. D Indicator functions as at (A3.3.4), and linear combinations of them, can be used to show that the :F-predictable a-algebra'¥~ above can be characterized as the a-algebra generated by the class of bounded left-continuous :F-adapted processes [e.g., see Lemma 3.1.1 of Kallianpur (1980)]. It is often important to examine the behaviour of a process not at a fixed timet but rather a random time T = T(w). Here the definition of a stopping time is fundamental. Definition A3.3.11. Given a history :F, a nonnegative r.v. T: n-+ [0, oo] is an :F-stopping time if {co: T(w)::::;; t} e 3"; (0::::;; t < oo). If S, T are stopping times, then so are S " T and S v T. Indeed, given a family { T,: n = 1, 2, ... } of stopping times, supn~ 1 T, is an :F-stopping time, while infn~ 1 T, is an ~+ 1 -stopping time. Since {T(w) = 00} = nn {T(w) > n} E 31'00, we can also consider extended stopping times as those for which &l{T(w) < oo} < 1. While stopping times can be generated in various ways, the most common A3.3. Processes and Stopping Times 651 method is as a first passage time, which for a nondecreasing process usually arises as a level crossing time. Lemma A3.3.111. Let X be an :F-adapted monotonic increasing right-continuous process and let Y be an ff0 -measurable r.v. Then T(w) =inf{ t: X(t, w) ~ Y(w)} is an :F-stopping time, possibly extended, while if X is :F-predictable then Tis an (extended) ~->-stopping time. PROOF. If Yis constant, X(t) ~ Y if and only if T::;; t, and since {w: X(t, w) ~ Y} E ff,, we also have {T(w)::;; t} E ff,. More generally, X(t, w)- Y(w) is monotonic increasing, right-continuous, and :F-adapted (because Y, being :F0 -measurable, is necessarily ff,-measurable for every t > 0). Then by the same argument, {T(w)::;; t} = {w: X(t, w)- Y(w) ~ 0} E ff,. Finally, when X is :F-predictable, it is ~-radapted, and thus we can replace ff, by ff,_ throughout. D The next result shows that a process stopped at an :F-stopping time T inherits some of the regularity properties of the original process. Here we use the notation (t ::;; T), X(t A T) = {X(t) X(T) (t > T). Proposition A3.3.IV. Let :F be a history, Tan :F-stopping time, and X a process. Then X(t A T) is measurable, :F-progressive, or :F-predictable, according to whether X(t) itself is measurable, :F-progressive, or :F-predictable. In all these cases, if T < oo a.s., then X(T) is an ff00 -measurable r.v. PROOF. The product IT-algebra ~(IR+) ® lK is generated by sets of the form (a, oo) x B for real finite a and B E lK. Since {(t, w): (t A T(w), w) E (a, oo) x B} =(a, oo) x (B n {T(w) >a}) and B n {T(w) >a} ElK, if X is measurable, so is Y(t, w) =X(t A T(w), w). The :F-predictable IT-algebra '¥~ is generated by sets of a similar product form but with BE ff... Since {T(w) >a} E ff.,, (a, oo) x (B n {T(w) >a}) is also a set generating'¥~, and thus if X is :F-predictable, so is Y as before. Suppose now that X is :F-progressive, so that for given t in 0 < t < oo, {X(s, w): 0 < s::;; t} is measurable as a process on (0, t] with probability space (Q, ff,, &>).Then the first argument shows that Y(s) =X(s A T) is a measur• able process on this space, that is, X(t A T) is :F-progressive. On the set {T < oo }, X(t A T) ~ X(T) as t ~ oo, so provided &>{T < oo} = 1, X(T) is a r.v. as asserted. D As an important corollary to this result, observe that if X is :F-progressive and a.s. integrable on finite intervals, then 652 APPENDIX 3. Conditional Expectations, Stopping Times, and Martingales Y(t, w) = J: X(s, w)ds is ~-progressive, Y(T) is a r.v. if T < oo a.s., and Y(t A T) is again ~ progressive. We conclude this section with some remarks about the possibility of a converse to Lemma A3.3.1. In the case of a quite general history, no such result of this kind holds, as is shown by the discussion in Dellacherie and Meyer (1978), especially around Chapter IV, Section 97. On the other hand, it is shown in the same reference that when X is defined on the canonical measure space (.il[o, ool• gJ(Jl[o, ool)) the two concepts of being ~-)-adapted and ~ predictable can be identified, a fact exploited in the treatment by Jacobsen (1982). The situation can be illustrated further by the two indicator processes Vi(t, w) =/{T(ro),;t}(t, w), Vi(t, w) =/{T(ro) 00 {w: Vr-(t) = 1} = {w: T(w) < t} = U {w: T(w)::::;; t- 1/n} E ~- n=1 Hence, both Vr+ and Vi are ~-progressively measurable [see the earlier comments or Bremaud (1981), Theorem A1.T33)]. Being left-continuous, Vi is ~-predictable [e.g., Theorem 1.T9 ofBremaud (1981)] and hence also ~->-adapted. No such statement can be made in general about Vi. However, suppose further that Tis not only an~ -stopping time but also an ~->-stopping time, so that from the above Vi is ~-> adapted. Can we assert that it is ~-predictable? Suppose that Tis a countably-valued r.v.; that is, 00 00 T-1 ( {tk: k = 1, 2, ... }) = U T-1(tk) = U Uk, say,= Q for some tk ~ 0. k=1 k=1 Then 00 {(t, w): Vi(t, w) = 1} = U [tk, oo) x Uk. k=1 By assumption, Tbeing an ~->-stopping time, Uk E ~k-• so Uk E u{U"~k-1/n} and hence vT+ is ~-predictable. While it can be proved that any ~-stopping time can be approximated from above by a sequence of stopping times taking only a countable set of values, this is not enough to treat the general case-indeed, the counter• example considered by Dellacherie and Mayer is just of this indicator function type. A3.4. Martingales 653 A3.4. Martingales Definition A3.4.1. Let (0, Iff,&') be a probability space, IF a history on (0, Iff), and X(·)= {X(t): 0 ~ t < oo} a real-valued process adapted to §i and such that EIX(t)i < oo for 0 ~ t < oo. Then X is an IF-martingale if for 0 ~ s < t < oo, (A3.4.1) E(X(t)i~) = X(s) a.s.; it is an §i -submartingale if (A3.4.2) E(X(t)i~) ~ X(s) a.s., and it is an 5 -supermartingale if the reverse inequality at (A3.4.2) holds. Strictly, we should speak of X as a .o/'-5-martingale: mostly, it is enough to call it a martingale since both 81' and 5 are clear from the context. While the concept of a martingale had its origins in gambling strategies, it has come to play a dominant role in the modern theory of stochastic processes. In our text we need only a small number of the many striking results concern• ing martingales and their relatives, principally those connected with stopping times and the Doob-Meyer decomposition. An important example of a martingale is formed from an §'00-measurable r.v. X00 with finite mean by taking successive conditional expectations with respect to §i: define (A3.4.3) X(t) = E(Xool~). Such a martingale is uniformly integrable. The converse statement is also true [e.g., Theorem 3.6 of Liptser and Shiryayev (1977)]. Proposition A3.4.11. Let X ( ·) be a uniformly integrable 5 -martingale. Then there exists on § 00-measurable r.v. X00 such that (A3.4.3) holds. The following form of the well-known convergence theorem can be found at Theorem 3.3 of Liptser and Shiryayev. Theorem A3.4.111. Let X ( ·) be an 5 -submartingale with a.s. right-continuous trajectories. If sup 0 ~t If also X(·) is uniformly integrable, then EIXoo I < oo and EIX(t)- Xoo 1--+ 0 as t--+ oo; that is, X(t)--+ Xoo in L 1 norm. This theorem can be applied to the example at (A3.4.3) whether the family of a-algebras {ff,} is increasing (as with a history 5) or decreasing. 654 APPENDIX 3. Conditional Expectations, Stopping Times, and Martingales For convenience, we state the result in terms of a two-sided history <£1 = {~t: -00 < t < 00 }, defining ~ Corollary A3.4.IV. If the r.v. Y is ~00 -measurable, has finite first moment, and Y(t) =E(YI~r) has a.s. right-continuous trajectories on -oo < t < oo for some two-sided history~. then (t--+ oo) (A3.4.4) (t--+ -oo) a.s. and in L 1 norm. In most point process applications, the processes concerned are right• continuous by definition, so the sample-path conditions for the convergence results above are automatically satisfied. In the general theory of processes, it is shown that, if the history .fF is right-continuous and the a-algebras are &'-complete in the strong sense that ffo (and hence ff, for all t > 0) contains all &'-null sets from J/'00 , there always exists a right-continuous modification of an .fF -submartingale, with the additional property that this modification also has left limits at each t > 0; that is, the (modified) process is cadhig [e.g., see Liptster and Shiryayev (1977, pp. 55-59) or Dellacherie and Meyer (1980); Elliott (1982) uses corlol, the acronym of the English equivalent, continuous on right, limits on left]. In turning to properties of martingales with fixed times s, t replaced by stopping times S, T say, we need the notion of u-algebras consisting of events prior to (and including) the time T and also strictly prior to T. Definition A3.4.V. Let .fF be a history and Tan .fi'-stopping time. The T-prior u-algebra J/'T is the sub-a-algebra of J/'00 J/'T ={A: A E J/'00 and An {T ~ t} E ff, for every t}, and the strict T-prior a-algebra J/'T- is generated by the sets {A: A E J/'0 } u {An {T > t} for A E ff, and fort:?: 0}. Clearly, J/'T and J/'T- are somewhat different entities [cf. Dellacherie and Meyer (1978, p. 117)]. It can be checked that Tis both J/'T- and J/'T-• measurable. A contrast is provided in the next result. Lemma A3.4.VI. Let .fF be a history, Tan .fi'-stopping time, and X(·) an .fi'-progressive process. Then X(T) is J/'T-measurable. Furthermore, if X(·) is .fi'-predictable, then X(T) is ff'T_-measurable. PROOF. Suppose X ( ·) is .fF -progressive. Setting for any x E IR, Ax= {m: X(T(m), m) ~ x}, A3.4. Martingales 655 X(T) is ~T-measurable if Ax n {T ~ t} E ~- But from Proposition A3.3.IV, X(t 1\ T) is ~-progressive, and therefore ~-adapted, so that {w: X(t 1\ T(w), w) ~ x} E ~(all x); hence, Axn {T ~ t} = {w: X(t 1\ T(w), w) ~ x} n {T ~ t} E ~- Now suppose that X(·) is ~-predictable. To show the ~T--measurability of X(T), look at the inverse image under X(T): w-+ X(T(w), w) E IR of a generating set (t, oo) x A {AE~) of the ~-predictable a-algebra '¥.?, namely, {w: t < T(w) < oo} n {w: wE A}, which is a generating set for ~T-. D The optional sampling theorem for martingales follows [e.g., see Liptser and Shiryayev (1977, pp. 60-61)]. Theorem A3.4.VII. Let ~ be a history, S and T the ~-stopping times with S ~ T a.s., and X(·) an ~-submartingale that is uniformly integrable and has right-continuous trajectories. Then fFs ~ ~T and E(X(T)ifFs) ~ X(S) a.s., where equality holds if X is an ~-martingale. Corollary A3.4.VIII. Let T be an ~-stopping time. If X(·) is a uniformly integrable ~-martingale (resp. submartingale) then so is X(t 1\ T). PROOF. For fixed s, t with s < t, s 1\ T and t 1\ T are two stopping times satisfying the conditions of the theorem, so E(X(t 1\ T)I~A T) ~ X(s 1\ T), and thus {X(t" T)} is a {~AT }-martingale. To show the stronger property that it is an ~-martingale, note that ~AT ~ ~ so {X (t 1\ T)} is ~-adapted, and it remains to show that (A3.4.5) LxtATPJ(dw) ~ Lx.ATPJ(dw) (all A E ~). knowing that it holds for all A E ~AT· Express the left-hand side as the sum ofintegralsover A1 =An {T > s} andA2 =An {T ~ s}.Certainly,A1 E ~. while A1 n {s 1\ T ~ u} =An {T > s} n {s 1\ T ~ u}. This equals 0 E ~ for u < s, and for u ~ s it equals A1 E ~ ~ ~. so by definition of ~AT• we have A1 E ~AT• and (A3.4.5) holds for A1 • On A 2 , 656 APPENDIX 3. Conditional Expectations, Stopping Times, and Martingales t 2': s 2': T so X(t 1\ T) = X(s A T) there, and (A3.4.5) holds for A 2 • By addi• tion, we have shown (A3.4.5). D Finally, we quote the form of the Doob-Meyer decomposition theorem used in Section 13.2, referring the reader to Liptser and Shiryayev (1977), for example, for proof. Theorem A3.4.IX. Let $' be a history and X ( ·) a bounded .fF -submartingale with right-continuous trajectories. Then there exists a unique (up to equiv• alence) uniformly integrable $'-martingale Y( ·) and a unique .fF -predictable cumulative process A(·) such that (A3.4.6) X(t) = Y(t) + A(t). For nondecreasing processes A(·) with right-continuous trajectories, it can be shown that $'-predictability is equivalent to the property that for every bounded $'-martingale Z( ·)and positive u, E f Z(t)A(dt) = E f Z(t- )A(dt). Since for any $'-adapted cumulative process ~ and any $'-martingale Z, E[Z(u) J~ ~(dt)] = E[J~ Z(t)~(dt)], the above property is equivalent to E[Z(u)A(u)] = E I: Z(t- )A(dt). A cumulative process with this property is referred to in many texts as a natural increasing process. The theorem can then be paraphrased thus: every bounded submartingale has a unique decomposition into the sum of a uniformly integrable martingale and a natural increasing function. The relation between natural increasing and predictable processes is discussed in Dellacherie and Meyer (1980). The boundedness condition in Theorem A3.4.IX is much stronger than is really necessary, and it is a special case of of Liptser and Shiryayev (1977)'s "Class D" condition for supermartingales, namely, that the family {X(T)} is uniformly integrable for all $'-stopping times. More general results, of which the decomposition for point processes described in Chapter 13 is in fact a special case, relax the boundedness or uniform integrability conditions but weaken the conclusion by requiring Y( ·) to be only a local martingale [i.e., the stopped processes Y( · 1\ T,) are martingales for a suitable increasing sequence {T,} of $'-stopping times]. The Doob-Meyer theorem is commonly stated for supermartingales, in which case the natural increasing function should be subtracted from the martingale term, not added to it. Given an $'-martingale X, it is square integrable on [0, r] for some r:::;; oo 2 ifsup0 [0, r]. When it is a bounded submartingale, the Doob-Meyer theorem as quoted above implies that we have the decomposition (A3.4.7) (0 s; t s; r) for some §"-martingale Y2 (") and §"-predictable process A2 (·). It is readily checked that for 0 s; s < t s; r, A2(t)- A2(s) = E((Xr- X.fl~), hence, the name quadratic variation process for A 2 ( · ). 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[428] Subject Index Absolute continuity of measures, 601 weak convergence of point processes, Actuarial mathematics, 5 277 Adapted stochastic process, 649 Additive set function, 596 Aftereffects, process with limited or Backward recurrence time r.v., 8, 19, 73 without, 13 distribution, 56 Almost sure convergence, 643 joint with forward recurrence time, Analytic orderliness, implies no batches 73 a.s., 29 Wold process, 91 Arbitrary points of point process, 42 internal history compensator, 562 Palm distribution interpretation, 484 Poisson process, 19 Arbitrary times for point process, 42 renewal process, 73 stationary point process interpretation, stationary point process, 56, 73 484 weak and strong convergence in Wold Atoms of random measure, fixed and ran• process, 98 dom, 172 Backward tree in cluster process, 310 Atomic measure, 610 Baire sets, 607 Autoregressive process for intervals, Bartlett spectrum of point process or ran- first order exponential, 89 dom measure, 411 infinitely divisible, 100 -for named processes, Autoregressive representation of random Bartlett-Lewis process, 417 measure, 436 bivariate Poisson process, 442 Averaging set in !Rd, see Convex averag• cluster process, 413 ing sequence Cox process, 416 Avoidance function, 132, 215 Gauss-Poisson process, 418 characterization by completely mono• Hawkes (self-exciting) process, 414 tone set function, 218 isotropic planar point process, 450 determines distribution of simple point N eyman-Scott process, 451 process, 216 multivariate random measure, 441 doubly stochastic Poisson process, 220 mutually exciting point process, 443 stationarity, implies stationarity of sim• Neyman-Scott process, 417, 451 ple point process, 319 Poisson process, 412 680 Subject Index Bartlett spectrum of point process or ran• Bonferroni inequalities, 120 dom measure (cont.) in p.g.fl. series expansion, 223, 232 quadratic random measure, 416 Borel measure, 598 renewal process, 412 boundedly finite, 154 Bartlett spectrum-general properties Borel sets, 598 canonical factorization, 432 cf. Borelian sets, 607 characterization, 414, 429 Borel a-algebra, 607 effect of random translations, 416 Boson process, 265 iterated, 416 discrete analogue, 268 effect of superposition, 416 likelihood, 501 existence of wide sense random meas- Boundedly finite Borel measure, 154 ure, 421 Boundedly finite measures on c.s.m.s., nonergodic random measure, 418 627 rational spectral density, 414, 433, 439 as metric space, 627 wide sense second order random meas- c.s.m.s. properties of, 629 ure, 426 weak convergence and equivalent topol• characterization, 429 ogies, 628 Bartlett-Lewis process, 246 Boundedly finite property, 39 Bartlett spectrum, 417 Poisson process, 25, 32 Batch-size distribution, 27, 48 stopping times have no finite cluster compound Poisson process, 50 point, 515 crudely stationary point process, 48 Branching process, 8, 9, 15, 142 moment as limit, 48 age-dependent, 8, 152 multiple points of sample paths, 48 critical, in iterated clustering, 308 Poisson process, 27 in mutually exciting process, 443 vanishing rth factorial moment meas• representation of cluster process, 324 ure, 233 sib distribution and length-biased sam- Bessel transform, isotropic planar point pling, 12 process, 449 use ofp.g.fl., 143, 150 Binning, 423 Binomial lattice process, Papangelou ker• nel, 586 Birth process, 9, 545 C.s.m.g. (complete separable metric birth rate estimator, asymptotic distri• group), 632 bution, 546 C.s.m.s. (complete separable metric conditional intensity, 546 space), 121, 595 Bivariate exponential distribution, planar metric space properties of a c.s.m.s. of random walk, 68, 71 boundedly finite measures on, 629 Bivariate point process, 249 equivalent topologies for a c.s.m.s. of not necessarily marked point process, totally finite measures on, 623 205 cadlag, 654 Palm-Khinchin equation, 384 Campbell equations for shot noise, 264 Poisson, 249 Campbell measure, 454 Bartlett spectrum, 442 characterization as measure, 460 noninfinitely divisible example, 250, conditions for stationarity, 461 254 compound, 581, 587 regular infinitely divisible, charac• discrete point process, 460 terization, 249 for any a-finite measure, 454 simple stationary, 384 uniqueness, 454 Blackwell's renewal theorem, 80 higher order, 455 generalized, 488 modified, 578 Wold process analogue, 95, 98 of KLM measure, 454, 459 Bochner's theorem, 403, 637 origin of terminology, 455 Bolzano-Weierstrass property, 595 Palm measure, 463 Subject Index 681 Radon-Nikodym approach to Palm the- mixing properties, 347, 353 ory, 455 moment measures, 239, 242 reduced, 478 via moments of cluster centre and refinement of moment measure, 455 member, 239 relation to first moment measure, 455 representation, second-order, 459 random thinning, 240 support, 469 random translation, 241 uniqueness, 454 stationary, 323, 329 Campbell theorem, 188 sufficient conditions, 329 Campbell-type measure for cumulative not necessary, 329 process, 524 variance/mean ratio, 366 in characterization of compensator, 525 variance function, 373 Canonical ensemble, Gibbs process, 125 Cluster random measure, 237, 242 Cartesian product, 602 Coherence in multivariate process, 442 Cauchy distribution as lifetime for ran• Coincidence density, 133; see also Fac- dom walk, 68 torial moment density Cauchy functional equation, 61, 329, 396 Compact regular measure, 611 Haar measure, 633 Compact space, 595 Central limit theorem, 4 locally, 596 functional, for random measures, 279 Compensator of point process and renewal processes, 69 cumulative process, 515 Centre-satellite process, 245; see also absolute continuity, 521 Neyman-Scott process characterizations, by Campbell-type Centred isotropic Poisson process, 380 measure and predictability, 525 Characteristic functional of point process conditional intensity function as deriva• or random measure, 14, 184 tive, 521 characterization, 186 continuity implies Cox process continuity properties, 194 representation, 549 moment measures as Fn!chet deriva• deterministic, implies process Poisson, tives, 191 550 of random linear functional, 184 for process with density, 516 Taylor series expansion, 190 IHF representation, 519 Characterization of Poisson process one-point process, 516 amongst orderly renewal processes with prior a-algebra, 518 infinite divisibility, 79 renewal process, 523 recurrence time r.v.s, 74 uniqueness, 522 superposition, 76 Compensators, convergence implies thinning, 75 processes converge weakly, 552 by complete randomness, 25 Complete covariance density, 364 by Poisson distribution, 29 Complete independence, 25; see also Characters of group, 637 Completely random dual group, 638 marked point process, 205 Chebyshev's inequality, 643 Complete intensity function, 505, 559 Cluster centre process, 236 existence for simple stationary point Cluster component or member process, process, 561, 563 236 Hawkes process, 559 Cluster process, 11, 236 hazard function representation, 562 Bartlett spectrum, 413 in convergence to equilibrium of branching process representation, 324 Hawkes process, 492 condition to be well-defined, 238 renewal process with density, 560 ergodicity, 347 Wold process, 560 ergodic theorem, 361 Complete separable metric space, see factorial cumulant measure, 372 C.s.m.s. independent, 237 Completely monotone set function, 217 682 Subject Index Completely monotone set function (cont.) Continuity, characterization of avoidance function, characteristic functional, 194 218 extended Laplace functional, 192, 195 Laplace transform of process directing p.g.fl.s and extended p.g.fl.s, 232 Cox process, 220 Continuity lemma, for measures, 597 Completely random measure, 158 Continuity set, stochastic, 270 consistency conditions satisfied, 181 Continuous mapping theorem, 618 Laplace transform, 180 Controlled thinning, 556 moments, 189 Convergence representation, 177 in variation norm, 272, 615 sample paths and components, 176 randomly thinned point process, 296 Completely random property, 25, 158, recurrence times in Wold process, 99 175, 205 renewal process, 80 Complex measure, 596 of conditional distributions, see Stable Compound Campbell measure, 581, 587 convergence Compound Poisson distribution, 11 of KLM measures, 282 cluster interpretation, generalized to of measures, 615 Khinchin measure, 145 in variation norm, 615 "doubly Poisson;' 121 of moment measures, 280 terminology, 10 ofr.v.s, Compound Poisson process, 24, 27 almost sure, 643 p.g.fl., 226 in distribution, 644 Concentration function, 301 in Lp norm or pth mean, 643 Condition l:, 578 in probability, 643 processes violating, 585 stable, 644 Conditional distribution, 640 of random measures and point regular, 641 processes, Conditional expectation, 639 fidi distributions, 271 predictable version, of process, 549 equivalent to weak, 272 Conditional intensity function for regular strong, 272 point process, 503 weak, 271 derivative of compensator, 521 to equilibrium, 484 Cox process estimator, 533 mixing properties, 344, 353, 488 left-continuous version used, 504 Convergence-determining class, 617 linear parameterization, 506 Convex averaging sequence, 332, 339 log linear parameterization, 506, 512 Corlol, see Cadlag renewal process, 512 Count spectrum of point process, 399 simple stationary point process on IR, Countably additive set function, 596 563 Counter process, Lampard's reversible, Conditional intensity, relative, 365 103 Conditional Janossy densities, 575 Counting measure, 8, 38, 197 Conditional probabilities, 604 metric space properties, 199 regular, 604 simple, 197 mixture of, 604 support, 198 Conditional probabilities, existence of Coupling arguments, 80, 490 regular, 166 Blackwell's renewal theorem, 80 Conditional risk function, see Condi• convergence to equilibrium of Hawkes tional intensity function, 504 process, 491 Configurations in Poisson process, 337 Covariance density, 364, 368, 378 ergodic theorem, 337 complete, 364 Consistency conditions, of random measure, 191 distribution of random measure, 168 Covariance measure, 357 Kolmogorov, 168 Hawkes process, 368 Subject Index 683 of random measure, 191 joint distribution with recurrence time, vehicular traffic process, 378 100,483 Covering ring, 613 Cyclic Poisson process, 24 semiring, 615 likelihood, 512 Cox process ( = doubly stochastic Poisson maximum likelihood estimation of process), 261 parameters, 513 avoidance function, 220 Cylinder set, 602 Bartlett spectrum, 416 measurable, 605 characterization of stationary renewal processes forming, 267 conditional intensity estimator, 532 directed by random measure, 262 Davidson's conjecture for line processes, Markov chain, 532, 547, 548 385, 395 Markovian diffusion, 540 developments from, 558 g:.compensator if point process sim- Dead time in interacting processes, 369 ple, 549 Decomposition of sample paths of ran- ergodic and mixing properties, 346 dom measure, 174 limit of thinned point processes, 292 Delayed renewal process, 71 Mecke's characterization, 300 Deny's theorem, renewal equation, 70 moment measures, 262 Derivatives of generating functionals, on cylinder for line process, 394 196, 224 p.g.fl. representation, 262 Determining class for set function, 596 point process with continuous compen• random measure distribution, 167 sator, 549 Deterministic component of random stationary iff directing process station• measure, 182 ary, 329 Deterministic point process, 74 weak limit of processes with convergent Bartlett spectrum, 412 compensators, 552 renewal process, 74 Cramer spectral representation, see Spec• stationary version, 329 tral representation, 418 Deterministic second-order random meas• Cross intensity, 370 ure, 430, 432 marked point process, 376 Diagonal factorization of second order Crude stationarity, 41 moment measure, 355, 371 batch-size distribution, 46 line process, 392 orderliness implies simplicity, 46 planar point process, 382 Poisson process, 25 Differentiation of Laplace functional, 196 Cumulant generating function, Taylor ser• Diffuse measure, 610 ies expansion, 114 Dirac measure, 173, 610 Cumulant measures, effect of shift operator on, 317 of random measure, 191 Direct Riemann integrability, 82 relation to moment measures, 192 conditions for, 88 Cumulative process martingale, 522, Wold process, 96 530 Directional rose Cumulative process on IR., 514 line process, 387 compensator, 515 moments of planar process, with density, 516 first order, 379 counting process, 514 second order, 381 history and internal history, 514 Neyman-Scott process, 384 intensity, as compensator derivative, Dirichlet process, 164 525, 530 consistency of fidi distributions, 171 natural increasing process, 656 moment measure, 195 Current lifetime of point process on IR, Discrete distribution, infinitely divisible, 73, 91 29 684 Subject Index Discrete time point process Ergodic limit of point process or random Campbell measure, 460 measure, 335 compensator and predictability proper- as invariant random measure, 360 ties, 526 ergodic process, 341 Palm measure, 460 KLM measure, 350 Poisson process analogue, 526 mean density, 336 Wold process, 91, 101 Laplace functional conditions for, 344 periodicity, 92 Ergodic point process, 58, 341 Disintegration, of kth order moment cluster process, 347 measure, 371 Cox process, 346 of measure invariant under a-group of iff metrically transitive, 342 transformations, 635 infinitely divisible point process, 350 Disks in plane, random measure defined invariant a-algebra characterization, by, 164 342 Dissecting ring, 609 Ergodic random measure, 341 Dissecting system, 172, 608 Palm measure, 465 in approximating point process entropy, Ergodic theorem, 330 566, 571 classical individual and statistical, 330 Distribution, spread out, 85 configurations in Poisson process, 337 Dobrushin's lemma, 45, 210 for point processes and random meas- Dominated convergence theorem, 600 ures, 335 Doob representation theorem, 642 kth order, 360 Doob-Meyer decomposition theorem, ~ version, 340 654 marked point process, 339 Doubly Poisson compound Poisson distri• Poisson cluster process, 361 bution, 121 shifts, 330, 333 Doubly stochastic Poisson process, see weighted average version, 339 Cox process Essentially bounded r.v., 643 Dual exvisible projection, 590 Exclusion probabilities, 122 Dual predictable projection, 525, 529 Existence theorem, Dynkin system, 593 random measure, 169 point process, 202 Expectation function U( ·) of stationary Edge correction for moment estimates, point process, 58 362, 374, 383 in ~ norm ergodic result, 340 Efficient score statistic, Expectation measure, renewal process, 64 Gauss-Poisson process, 513 Exponential autoregressive process, Neyman-Scott process, 513 first order, 89 point process on JRd, 507 time reversed process of, 102 Poisson cluster process, 509 Exponential distribution order properties, Elementary renewal theorem, 70 7, 22 analogue for stationary point process, Exponential formula for integration by 58 parts, 103 compared with Blackwell's theorem, 80 in Cox process for point process with from strong law, 69 continuous compensator, 549 Engineering applications, 16, 485, 528 integral equation for likelihood ratio of Entropy increasing operations, 269 point processes, 535 Entropy of point process, 563 Extended Laplace functional, 192 maximum for Poisson, 565, 571 Extended p.g.fl., 222 Entropy rate of point process, 566, 569 convergence properties, 232 interval, 569 mixing property of cluster process, 348 renewal process, 572 Extension problem for product measures, of approximating renewal process, 572 605 Equidistant points, process of, 74 Kolmogorov theorem, 606 Subject Index 685 Extension theorem, determined by internal history, 107, distribution of random measure, 169 527,551 for finitely additive set function to of random measure, 166 measure, 597 Filtering problem, 531, 534, 538 problem for signed measures, 161, 598 Filtering theory, 495 Exterior conditioning, 574 Filtration for stochastic process, 649; see Exvisible a-algebra, 590 also History Finite dimensional distributions, see Fidi distributions Finite intersection property of c.s.m.s., Factorial cumulant measures, 146 595 generating functional, 149 Finite(= a.s. finite) point process, 109 higher order vanish for Poisson process, consistency of fidi distributions, 203 226 defined by finite vector, 201 in log p.g.fl. expansion, 224 entropy, 564 Factorial cumulants, 114 infinitely divisible, 256 linear for Poisson cluster process, 372 moment structure, 129 relation to factorial moments, 114 specification, 126 Factorial moment density, 133 Finite renewal process, 123 renewal process, 136 Finite vector, in formal definition of Factorial moment measures, 130 finite point process, 201 generating functional, 148 Finitely additive set function, 596 in p.g.fl. expansion, 222 extension to measure, 597 relation to Janossy measures, 131 First order stationarity, 355 Factorial power, 112 expectation measure characterization, expectation function and moment 356 measure, 130 Fixed atom Factorization, canonical, Poisson process, 32 Bartlett spectrum, 432 random measure, 172 orthogonal increment process, 433 Forward recurrence time, 19, 55 Wold decomposition, 432 as marks, 377 Factorization lemma, 634 distribution, first moment measures, convergence, 83 isotropic process, 379 joint, with lifetime, 483 line process, 387 stationary point process, 55 marked point process, 375 strong convergence, 88 Palm theory, 461 Poisson process, 19 second moment measures, renewal process, 66, 71, 490 isotropic centred process in plane, variation norm convergence, 491 382 weak convergence, 490 stationary isotropic line process, 392 Fourier methods, 4 stationary isotropic process in plane, Fourier transform, 636 382 function of rapid decay, 400 shift invariant measures, 333, 355, p.p.d. measure, 404 375, 385, 635 Poisson process, 423 Fatou's lemma, 600 reduced covariance measure, 411 Fermion process, 136 reduced second moment measure, 411 discrete analogue, 139 unbounded measure, 399 likelihood, 501 variance function, 411 reduction to renewal process, 140 Fourier-Stieltjes transform, 637 Fidi (finite dimensional) distributions, Frechet derivative identification of of point process, 202 moment measures, 191 consistency, in Poisson and finite Fubini's theorem, 603 point process, 203 Functions of bounded support, 220 686 Subject Index Functions of rapid decay, 400, 408, 429 in Fourier and inverse Fourier formu• dense in Hilbert space ~(r), 419, 428 lae, 638 uniqueness on Abelian group by normalization if compact, 633 Gamma distribution, 3 Cauchy functional equation solution, point process intervals with, 93, 100 633 simulation of Wold process, 102 Halo set, 611 Gamma random measure, 163 Hamel equation, 61; see also Cauchy condition to be nonatomic, 181 functional equation stationary, 158 Hard core point process model, 125, 366; satisfies consistency conditions, 170 see also Matern Gauss-Poisson process, 247 Harris recurrent Markov chain, 94 as limit of u.a.n. array, 289 Hawkes ( = self-exciting) process, 367 as pseudo-Cox process, 266 Bartlett spectrum, 414 Bartlett spectrum, 418 cluster process representation, 367 characterization, 248, 254 complete intensity function, 491, 505, efficient score statistic, 513 559 isotropic centred, 380 conditional intensity and likelihood, Janossy densities, 500, 511 505 likelihood, 500 convergence to equilibrium, 491 satisfying Condition I:, 586 covariance measure, 367 . two-dimensional, 383 infectivity function conditions for vari- Gaussian measures on Hilbert space, 184 ation norm convergence, 492 General renewal equation, 65 initial conditions, 492 General theory of processes, 515 effect in estimation, 505 Generalized entropy, 564 linear and optimal predictors coincide, point process, 565 505 Generalized function, spectral representa- moments, 367 tion, 418 multivariate, 443; see also Mutually Generalized Poisson, terminology, 10 exciting process Generalized random process, 184 optimal predictor linear, 560 Generating function interrelations, 116 self-exciting component, 492 Generating functional Hawkes self-exciting process, see Hawkes derivatives yield moment measures, process 224 Hazard function, 2; see also IHF factorial moment and cumulant meas- in conditional intensity for regular ures, 148 point process, 503 interrelations, 149 intervals of Poisson cluster process, Janossy measures, 148 245, 253 Khinchin measures, 149 nearest neighbour distribution analogue Geometric distribution, in thinning in IJld, 253 renewal process, 76 Neyman-Scott Poisson cluster process, Gibbs kernel, 581 253 Gibbs process, 124, 573 of interval distribution, 56 no interaction (ideal gas), 125 renewal process, 103 repulsive interaction, 125 Wold process, 512 Greig-Smith quadrat method, 366 Hazard measure, 106; see also IHF Group, 631 characterizes Wold process, 103 characters of, 637 extension from hazard function, 103 Heine-Bore! property, 595 Hermite distribution p.g.f., 121 Haar measure, 633; see also Cauchy Hilbert space, functional equation functions of rapid decay dense in ~(r), in factorization of measures, 634 419, 428 Subject Index 687 Gaussian measures on, 184 cluster process representation, 323 isomorphism between L2(r) and L2 (l;0 ), ergodicity and mixing properties, 419 350, 354 History, 514, 649 KLM measure, 323, 329, 350 for doubly stochastic processes, 515 Palm measure, 468 internal or minimal or natural, 649 factorization characterization, intrinsic, 649 473 prior to stopping time, 654 regular/singular classification, 323 right-continuous, 515 tail a-algebra, 353 Homing set, regenerative, for Markov weakly/strongly singular classifica• chain, 93 tion, 352 Homogeneous chaos, Wiener's, 161 weakly singular, from iterated Hyperplane process, 390, 396 cluster mechanism, 312 u.a.n. array condition, 284 Infinitely divisible random measure, l.i.d. clusters, 110, 122 Laplace functional characterization, p.g.fl.' 145 287 IHF (integrated hazard function), 106 stationary, KLM and Palm measures, analytic properties for simple point 468 process, 527 Inhibition, compensator property, 519 renewal process model, 373 one-one relation with lifetime d. f. , 107 Matern model, 367, 373 Independent cluster process, 237 Inhomogeneous Poisson process, see Independent random measures, 194 Nonstationary Poisson process Independent (triangular) array, 281 Integrated hazard function, see IHF Indistinguishable points in point process, Integration by parts for Lebes• 21 gue-Stieltjes integrals, 103 Individual ergodic theorem, 330 exponential formula, 103 for d-dimensional shifts, 333 Intensity Infectivity model, 151, 367; see also cross, 370 Hawkes process marked point process, 376 Infinite superposition of point processes, _g1:intensity of cumulative process, 525 230 existence of predictable version, 525, cluster process as, 237 530 Infinitely divisible, derivative of compensator, 530 autoregressive process, 100 of point process, linear prediction, 439 discrete distribution, 29 Papangelou, 575 distributions, convergence, 282 second order, 365 Infinitely divisible point process, 14, 255, stationary point process, 44 322; see also Poisson randomiza• Intensity function, tion complete, 559 a.s. finite, conditional, for regular point process, p.g.fl. characterization, 256 503 interpretation as Poisson randomi• dependent on history, 531 zation, 256 linear predictor for, 439 interpretation as Poisson cluster Intensity measure, 209; see also Com• process, 257 plete, Conditional, and Papangelou invariance under a-group and KLM intensity measure invariance, 329 Korolyuk's theorem, 45, 50, 210 p.g.fl. representation, 258 Interacting particle systems, 15, 124, 573 Poisson process, 259 Interacting point processes, 368, 374 regular/singular classification, 259 dead time in, 369 stationary, 322 Interaction potential, used to define point characterization, 473 process, 124, 129, 573 688 Subject Index Internal history, 514, 649 Iterated cluster formations, 308 canonical space for, 559 stable/unstable cluster mechanisms, fidi distributions determined by, 107, 310 527,551 Iterated random translations, 300 right-continuity, 515, 528 Bartlett spectrum, 416 Interval function, spectral representation, 418 Intervals of point process on IR, 38 Janossy density, 122, complete intensity function, 559 expansion via Khinchin densities, 501 correlated exponential r.v.s, 93 in exterior conditioning, 573 correlated gamma r.v.s, 93 local, as point process likelihood, 497 hazard function, 56 probabilistic interpretation, 122 measure-theoretic definition, 200, 207 renewal process, 123 mth order Markov chain, 103 Janossy measure, 122 one-one correspondence with counting defining point process distribution, 127 measures, 200, 207 generating functional for, 148 Ryll-Nardzewski-Slivnyak theorem, 14, local, 230 475 stationary point process on circle, 471 spectrum, 399 Jensen's inequality, 640 stationarity, 42 Jordan-Hahn decomposition, 598 Intrinsic history, 519, 649 Invariance of process under shifts, 318 under other transformations, 325 under rotations and shears, 386 Kallenberg's randomized lattice, 395 Invariant events, 331 Key renewal theorem, 80, 83; see also sample path relations for stationary Blackwell's renewal theorem and Palm distributions, 493 Breiman's form, 85, 96 stationary point process, 485 Khinchin density functions, Invariant random measure, from ergodic in nonstationary Poisson process likeli• limit, 360 hood, 507 Invariant a-algebra (a-field), 331, 340, Khinchin measures, 145 350, 361 generating functional for, 149 discrete vs. continuous time, 331 local, 231 relation to tail a-field, 343, 353 Khinchin orderliness, 49, 211 trivial, not equivalent to orderliness or simplic• characterizing ergodicity, 342 ity, 49 regular infinitely divisible distribu• Khinchin's existence theorem, 44 tion, 353 intensity measure version, 209 Inversion relation, KLM (= Kerstan, Lee, Matthes) meas- Fourier transform, 637, 638 ure, 258, 323, 329, 350, 354 Palm and stationary distributions, 475, Campbell measure, 454 480,484 ergodicity and mixing, 350 p.p.d. measures, 404 infinitely divisible point process spectral process Z('), 422 characterization, 260 Ising problem, 575 Poisson process, 259 Isotropic process, 365 Q-weak convergence, 282 line process, 385, 389, 393 stationary, 323, 329 stationary, 391 Kolmogorov consistency conditions, 168 planar, 379 Kolmogorov extension theorem, 606 centred Poisson, 380 for KLM measures, 261 centred Gauss-Poisson, 380 Korolyuk equation, 48 second order properties, 449 generalized, 210 stationary, Bartlett spectrum, 450 Korolyuk's theorem, 45, 50, 210 Subject Index 689 Laguerre polynomials, isotropic, 385, 389, 393 in likelihood estimation, 505 line measure, 388 in Wold process, 102 mean density, 388 Laplace functional of random measure, parallel and antiparallellines in, 391, 192 397 characterization of Palm distribution, parametric representations, 386, 396 457 p.g.fl.' 390 continuity properties, 192 Poisson, 390 superposition example, 288, 289 railway, 397 derivative, 196 Line process in IR 3, 397 extended, 192, 345 Line segments, random measure defined infinitely divisible random measure, by, 164 287 Linear filter, 422 ergodicity and mixing properties, 344 Linear functional of random measure, relation with p.g.fl. of point process, 182 221 broad and strict sense equivalence, 186 Laplace transform representation of com- Linear prediction, 429 pletely random measure, 180 intensity, 439 Lebesgue convergence theorems, 600 point process with rational spectral Lebesgue decomposition theorem, 601 density, 439 Lebesgue-Stieltjes integration by parts two-point Poisson cluster process, 440 formula, 104 Linear process, 196 Length-biased sampling, 13 Local Janossy density, distribution, 83 defines point process likelihood, 497 explanation of waiting time paradox, infinite particle system, 574 42, 55 Papangelou intensity as ratio, 576 Level crossing problems, 14, 294 Local Janossy measures, 230 Levy-Khinchin representation, 79 in p.g.fl. series expansion, 231 Life table, 2 Local Khinchin measures, 231 Lifetime of renewal process, 2, 64; see Local martingale, 516 also Current lifetime as integral with respect to martingale, Likelihood derivatives, asymptotic nor• 527 mality, 540 as structure of likelihood ratios, 536 Likelihood, 22, 497 Local Palm distribution, 456 boson process, 501 Papangelou intensity, 576 cyclic Poisson process, 512 Poisson process, 458 fermion process, 501 Locally compact space, 596 nonstationary (inhomogeneous) Poisson Log p.g.f. series expansion, 115 process, 497 compound Poisson distribution pairwise interaction system, 501 interpretation, 115 Poisson process, 20 Logarithmic distribution, 11 renewal and stationary renewal Lognormal distribution, 3 process, 512 Loss of memory, 23 Wold process, 102, 512 convergence to equilibrium, 484 Likelihood ratio, 498, 534 integral equation for, given predictable intensities, 535 MacMillan's entropy rate theorem, 566 local martingale structure, 536 Marginal probability measure, 603 Line measure, for line process, 388 Marked point process, 204 Line process in plane, 386 completely independent, 205 directional rose, 387 convergence to, 286 factorization of second moment meas• in random thinning, 294 ure, 392 cross intensity, 376 690 Subject Index Marked point process (cont.) Measure on product space, ergodic theorems, 339 factorization of, invariant in one com• Palm distributions, 495 ponent, 634 forward recurrence times as marks, Measure, 596 377 atomic, 610 independent marks, 376 Borel, 598 infinitely divisible, 261 boundedly finite, 627 random measure defined by, 163, 214 compact regular, 611 representation of moment measure, complex, 596 375 diffuse, 610 stationary, 375 Dirac, 173, 317, 610 moment measure factorizations, 375 Haar, 633 zero/infinity property, 385 invariant under a-group of transforma- Markov chain, tions, 635 defmes intervals of point process, 203, positive-definite, 401 206 p.p.d. (positive positive-definite), 401 discrete, Radon, 631 periodicity and Wold process, 92 regular, 610 strongly aperiodic recurrent, 94 signed, 596 Harris recurrent, 94 tempered, 409 point process with intervals from, 89 tight, 611 splitting technique and Wold process, totally finite, 597 93 transformable, 401 Markov chain transition kernel, translation bounded, 401, 410, 429 bivariate expansion for, 92 Mellin transform, use in Wold process, power diagonal expansion, 101 102 strongly aperiodic recurrent, 94 Method of backward trees, 310 Wold process, 90 Method of reduced trees, 313 Markov shifts of point process, invariant Metric compactness theorem, 595 measures for, 315; see also Ran• Metric on topological space, 594 dom translation Metric spaces, equivalent, 595 Martingale, 653 Metrically transitive transformation, 331 for cumulative process, 522 ergodicity equivalence, 342 from cumulative process with predict• point process and random measure, able .9"-intensity, 530 342 from point process with predictable .9"• Minimal history, see Internal history intensity, 540 Mixed Poisson distribution, 10 square integrable, 656 terminology, 10 Martingale convergence theorem, 653 Mixed Poisson process, 24 Matern hard core process, 366, 373 asymptotic normality of rate parameter Maximum likelihood estimation in Pois- estimate, 544 son process, 513 generalization to Cox process, 262 Mean density, intensities under different histories, as ergodic limit, 336 531 intersection of lines in plane, 397 mixing iff simple Poisson, 347 line process and line measure, 388 p.g.fl., 227 pairs of points, 363 stationary, Poisson process, 19 invariant under random translation, stationary point process, 43, 356 306 Mean square integral, 420 orderliness properties and counter Measurable family of point processes, examples, 49, 214 235 Palm measure, 467 Measurable space and function, 599 particles with random velocities, Measurable stochastic process, 648 307 Subject Index 691 Mixing of point process or random meas- Multiplicity, in Poisson process, 27 ure, 341 Multivariate point process, 205 convergence to equilibrium, 488 asymptotic second moment properties, Cox process, 346 493 cluster process, 347, 353 infinitely divisible, 261 infinitely divisible point process, 350, Poisson, convergence to, 286 354 Multivariate random measure, Laplace and p.g.fl. formulation, 344 coherence and phase, 442 mixed Poisson process, 347 spectral measure, 441 of order k, 354 Mutually exciting point process, Bartlett Palm distribution convergence, 487 spectrum, 443 renewal process, 352 tail a-algebra, 343 weakly, 341 Natural history, see Internal history Mixture of conditional probabilities, 604 Natural increasing process, 656 Mixture of random measures, Palm distri• Nearest neighbour distance, 16 bution, 468 Poisson cluster process in IR, 247 Modification of stochastic process, 649 Poisson process in R•, 23 Modified Campbell measure, 578 stationary process in R•, 253, 479 absolute continuity, 579 Negative binomial distribution, 10, 12 Modified renewal process, 71; see also compound Poisson process, 228 Delayed renewal process expansions of p.g.f., 115 Modulus of continuity, 99 mixed Poisson process, 229 in renewal theorem analogue, 99 Negative binomial process, 228 Moment density, 70; see also Product p.g.fl., 231 (moment) density Nested partitions, dissecting system, 172, Moment generating function, p.g.f. and 608 circles of convergence, 120 Neyman-Scott process, 245 Moment measure of point process or ran• Bartlett spectrum, 417, 451 dom measure, 130, 188 Cox process directed by Markov chain asymptotic behaviour on convex aver• model, 547 aging sequence, 360 efficient score statistic, 513 estimation, 363 in IR2, 252, 373, 383 edge correction in, 362, 374, 383 directional rose, 384 disintegration of kth order, 371 isotropic, Bartlett spectrum, 451 factorization when stationary, 356 radial correlation function, 373 diagonal disintegration, 372 regular representation, 324 line process, 392 Ripley's K-function, 384 planar point process, 382 shot noise process, 264 generating functional derivative, 224 stable cluster process from iteration, higher order determine lower order, 311 139 Nonatomic (= diffuse) random measure, reduced, 357 174 relation to cumulant measures, 149, moment measure condition for, 196 192 Nondecreasing integer-valued step• stationary, 355 function defining point process, 38 Moment measures, convergence, 280 Nondeterministic random measure, 430 Monotone class theorem, 593 Nonergodic random measure, Bartlett Monotone convergence theorem, 600 spectrum, 418 IJ.-orderly point process, 211 Nonnegative sequence, Multiple points of sample paths, 48 measure-theoretic definition of point point process without, 48 process by, 201 Multiplicative population chain, 142; see Markov chain, 206 also Branching process random walk, 207 692 Subject Index Nonstationary (= inhomogeneous) Pois• local, 456 son process, 21 regular family, 456, 560 likelihood, 497 moments, 464 Khinchin density functions, 507 of random measure with density, 456 likelihood ratio, 499 Palm kernel, 456 maximum likelihood estimation of Laplace functional relations, 368, 457 parameters, 513 Poisson process, 458 Null array, 281 Palm measure, see also Palm distribution, Palm-Khinchin equations as expectation of invariant random One-point process, 516 measure, 466 compensator, 517 as rate of marked point process, 472 with prior a-algebra, 518 convergence to stationary measure, with randomized hazard function, mixing condition for, 487 519 inversion relations with stationary dis- Optional sampling theorem, 655 tribution, 475, 480, 484 Ordered vs. unordered sets of points, of mixed Poisson process, 467 121-122 of point process on integers, 460 Orderliness properties, 28, 43, 207 of stationary ergodic random measure, analytic orderliness, 29 465 implies no batches a.s., 29 of stationary point process on circle, in mixed Poisson process, 49, 214 471 in Poisson process, 28, 33 of stationary random measure, 463 interrelations, counterexamples to, 49, of stationary simple point process on 214 IR, 474 Khinchin orderly, 49, 211 second-order, of stationary process, 1-1-orderly point process, 211 459 orderly point process, 28, 44 Palm probabilities, from ergodic theorem, relation to intensity, 215 337 simple, 28, 44, 197 Palm-Khinchin equations, 50, 475 without multiple points, 48 bivariate point process, 384 Ordinary, 210; see also Orderliness differential form, 54, 482 origin of term, 28, 210 Slivnyak extension, 57, 479, 483 necessarily simple, 211 Ryll-Nardzewski theorem, 475 Orthogonal increment process, random stationary point process on IR, 13, 50 measure from, 420, 422, 426, 434 heuristic derivation, 54 Overdispersion of count distributions, 9, stationary renewal process, 477 24 stationary Wold process, 478 Poisson cluster process, 244, 253 Palm's original derivation, 378 Cox process, 263 Papangelou intensity, 575 ratio of Janossy densities, 576 Papangelou intensity measure, 577, 587 Pairwise interaction system, likelihood, dual exvisible projection, 590 501 Papangelou kernel, 577 Palm distribution, see also Palm measure atomic component, 583 as ergodic limit of random shifts of sta• existence and uniqueness, 589 tionary distribution, 485 of higher order, 580 conditional probability interpretation, Parameter measure of Poisson process, 481 32 convergence in variation norm decomposition, 32 property, 484 randomized, defines Cox process, 261 existence of kth order moment, 464 in kth factorial moment measure of Laplace functional characterization, process, 226 457 Parseval-Bessel equation, 450 Subjectlndex 693 Parseval identity and relations, 637 Phase space of random measure, 154 Bartlett spectrum, 411 Planar point process, 378 extended, for Lebesgue integrable func- directional rose, 379, 381 tions, 405 factorization of second-order moment generalized functions, 400, 405 measure, 382 p.p.d. measures, 400 isotropic, 379 process with orthogonal increments, stationary, Bartlett spectrum, 450 434 rotational invariance, 378 random measures, 421, 422, 428 Plancherel identity, 636 Particle system, 124, 129 generalized, 638 finite vs. infinite, 57 4 Point process-basic properties and mis- equilibrium distribution, 574 cellaneous topics Particles with random velocities, mixed counting measure for, 8, 38 Poisson process, 307 defined by point set, 122 Partition function, Gibbs process, 124 definitions on ilt 38 Partitions, equidistant points in, 74 of set in state space, 172, 608 fidi distributions, 202 dissecting system defined by nested, informal definition, 18 172,608 measure-theoretic definition, 199 of sets of integers, 118 via finite vector, 201 Periodicity, Wold process, 92 via nonnegative sequence, 201 Periodogram, point process, 399 on state spaces other than IR Perturbation expansion, renewal function circle, stationary, Palm measure, 471 and spectral density, 93 cone, 385 P.g.f. (probability generating function), cylinder, line process representation, 10, 113 386 radius of convergence and m.g.f., 120 integers, see Discrete Taylor series expansion, 113 plane, see Planar P.g.fl. (probability generating functional), periodogram, 399 15, 141, 220 random measure characterization, 199 characterization, 221 random variable characterization, 200 of named processes sample path specifications, 38 cluster process, 237 transformed to Poisson process under compound Poisson process, 226 random time change, 550 conditional, for renewal process on Point process-classes or types of (see interval, 233 also Individual entries) Cox process, 262 cluster, 236 infinitely divisible point process, 258 i.i.d. clusters, 245; see also Ney• line process in IR 2, 390 man-Scott process mixed Poisson process, 227 correlated point pairs, 247; see also Poisson cluster process, 244, 250 Gauss-Poisson process Poisson process, 225 deterministic, 74 properties discrete, 92, 460 continuity and convergence, 231 finite(= a.s. finite), 109 of extended p.g.fl.s, 232 infinitely divisible, 255 determines point process distribution, isotropic, 365 222 marked, 204 ergodicity and mixing conditions, multivariate, 205 344 planar, 378 relation with Laplace functional, 221 rational spectral density, 214 superposition of independent point regular, 496 processes, 229 renewal, 64 Taylor series expansion, 222 simple, 44, 209 Phase of multivariate process, 442 stationary, 42, 316 694 Subject Index Point process-functions or properties of Point sequence, point process specified (see also individual entries) by, 38 backward and forward recurrence Poisson approximant of point process, times, 73 283 batch-size, 27, 48 convergence of superposition, 283 Bartlett spectrum, 411 Poisson binomial distribution, 297, 300 compensator, 515 in random thinning, 297 current lifetime, 73 Poisson branching process, 246; see also entropy and entropy rate, 563, 566 Bartlett-Lewis process ergodic, 58, 341 Poisson cluster process, 243 history, 514 a.s. finite clusters intensity, 44 p.g.fl. and charact !rization, 250 .9=intensity, 525 probabilistic interpretation, 254 intensity measure, 209 regular infinitely divisible process conditional intensity measure, 587 characterization, 260 intervals, 38, 200 regular representation, 252 likelihood and likelihood ratio, 22, 497 zero cluster not estimable, 252 linear predictor, 429 conditions for existence, 238, 243 martingale, 522 dependent clusters, 361 mean density, 19, 43, 356 efficient score statistic with bounded metrically transitive, 342 clusters, 509 mixing and weakly mixing, 341 factorial moment measures, 244 moments and moment measures, 130, infinitely divisible, 255 188 likelihood representation and Janossy orderliness, 43, 207 density, 511 Palm distributions and measures, 453 p.g.fl., 244 Palm-Khinchin equations, 50, 475 Poisson distribution, 5, 8 p.g.fl., 141, 220 Poisson process characterization, 30 second-moment (variance, covariance, Poisson line process, 390 correlation and autocorrelation) Poisson process, 18, 203 58, 365 asymptotic normality of rate parameter support process, 198 estimator, 544 spectral density, 414 Bartlett spectrum, 412 Point process-named (see also individual Fourier transform, 423 entries) batch size distribution, 27 Bartlett-Lewis, 246 bivariate, 384 Cox, 261 boundedly finite, 32 Gauss-Poisson, 247 centred isotropic, 380 Gibbs, 124, 573 characterization Hawkes (self-exciting), 367 by avoidance function, 31 Neyman-Scott, 245 by complete randomness, 25 Poisson, 18 by forward recurrence time amongst compound Poisson, 24 orderly renewal processes, 74 doubly stochastic Poisson, 261 by Poisson distribution, 30 mixed Poisson, 24 by renewal process property of super• quasi-Poisson, 30, 31 position, 76 Poisson cluster, 243 complete independence, 25 Wold, 89 condition for translation (Markov shift) Point process martingale, 522, 530, 540 invariance, 242 absolute continuity of compensator, 521 configurations in, 337 asymptotic mixed normality of consistency of fidi distributions, 203 integrals, 541 cyclic intensity function, 24 asymptotic normality of randomly parameter estimation, 24, 513 normed integrals, 543 discrete analogue, 526 Subject Index 695 isotropic centred, 380 measure, 401; see also P.p.d. measure KLM measure, 259 sequence, 409 lack-of-memory (without aftereffects) Positive positive-definite, see P.p.d. property, 13, 29, 66, 75 P.p.d. function, as covariance density, likelihood,20 418 limit of superpositions of u.a.n. array, P.p.d. (= positive positive-definite) meas• 285 ure, 401 mean density, 19 in IR 2, 409 mean rate, 19 inversion formula, 404 moment properties, 19 Predictability, of conditional expectation, multiplicity, 27 549 noise in telegraph signal, 528 with respect to internal history, 527 nonstationary (= inhomogeneous), 21 Predictable process, 515, 650 nonuniqueness ofp.g.f., 25, 252 Predictable a-algebra, 515 ordinary, 214 Prediction of point process, Palm kernel, 458 linear, 429, 560 characterization, 459 nonlinearity of optimal predictor, 560 local Palm distribution, 458 Previsible, 650, see Predictable parameter measure, 32 Probability distribution, spread out, 85 p.g.fl.' 225 Probability generating function, see P.g.f. random translation, 32 Probability generating functional, see randomly time changed point process, P.g.fl. 550 Product (moment) density, 15, 133; see renewal process with exponential inter- also Factorial moment density vals, 66 renewal process, 70 scale invariant, 325 Product topology and measure, 602 second moment, 70 Progressively measurable process, 649 simple if and only if ordinary, 214 Prohorov metric, 280, 622 simulation of nonstationary, 23 boundedly finite measures, 627 stationarity characterization, 319 on totally finite measures, 622 stationary, 18, 319 Prohorov's theorem, 619 invariant under i.i.d. random transla- for c.s.m.s. of boundedly finite meas• tion, 242 ures, 630 survivor function, 19 Proper difference of sets, 592 thinning nonstationary, 31 variance function, 373 Poisson randomization, 256 Quadratic random measure, 156 asymptotics of Palm measure of KLM Bartlett spectrum, 416 measure, 474 mixture, 189 in infinitely divisible point process moment measures, 189 characterizations sample path properties, 175 equivalent to totally finite KLM Quadrats, Greig-Smith method, 366 measure, 260 Quasi-Poisson process, 30, 31 if stationary then not ergodic, 349 not stationary, 319 strongly singular, 352 Queueing theory, 5, 452 if stationary then singular, 323, 474 Quotient topology, 633 infinite superposition with infinite KLM measure, 330 characterizes strong singularity, Radial correlation function, 365, 383 354 Neyman-Scott process, 373 Poisson summation formula, 402, 409 Radon measure, 631 Polish space, 595 Radon-Nikodym theorem, 601 Positive-definite, in Palm theory, 452 function, 401, 637 role of Campbell measure, 455 696 Subject Index Raikov's theorem, 76 on plane, spectral measure, 409, 450 Railway line process, 397 orthogonal increments, 420, 422, 426, Random deletion, see Random thinning 434 Random flats and hyperplanes, 390, 396 purely nondeterministic, 430 Random integral, 182 rational spectral density, 433 absolutely continuous, 195 scale invariant, 325 Random linear functional, 182 self-similar, 326 broad and strict sense, 183 signed, 162, 415 coincide if tight, 185 stationary gamma, 158 characteristic functional, 184 uniform, 156 Random measure-basic properties wide sense stationary, 415, 421, 424 characteristic functional, Random measure-functions or properties characterization, 186 (see also individual entries) Taylor series expansion, 190 autoregressive representation, 436 characterization of point process, 199 convergence, 271 constructed or defined by, covariance density, 191 conditional expectation of another covariance measure, 191 random measure, 161 cumulant measures, 191 line segments, 164 expectation measure, 188 disks in plane, 164 moment measures, 188 marked point process, 163, 214 absolutely continuous, 190 determining class, 167 first moment measure, 188 process with nonnegative increments, kth order, 188 157 second order stationary, decomposition, 174 spectral theory, 410 existence theorem, 160, 169 spectral representation, 418 fidi distributions, 166 stationary, consistency conditions, 168 Campbell measure characterization, extension theorem for, 169 461 measure requirements, 168 Palm measure, 463 fixed atoms, 172 Wold decomposition, 430 at most countably many, 173 Random measures, free of, 172, 174 independent, 194 measure-theoretic definition, 154 relatively weakly compact, 280 nonatomic, 174 superposition, 193, 281 moment measure condition for, 196 uniformly tight, 273 random variable characterization, 155 Random norming of estimators, 543 sample path properties, 171 Random probability distribution, 164 absolute continuity, 175 moment measures, 195 deterministic component, 182 Random sampling of random process, 426 state space, 154 Random Schwartz distribution, 184 Random measure-classes or types (see Random signed measure, 162 also individual entries) spectral measure, 415 chi-square density, 156 Random thinning of point process, cluster, 237, 242 as cluster operation, 240 completely random, 158, convergence in variation norm, 296 deterministic, 430, 432 dependent thinning, 554 differentiable, local Palm distribution, Cox process as limit, 556 456 limit, multivariate, 163 Cox process, 292 spectral measure, 441 Mecke's thinning characterization, on circle, spectral measure, 445 300 on integers, stationary Palm measure, mixed Poisson, 291 468 Poisson, 290 Subject Index 697 moments of thinned process, 240 Regeneration points, 13 of Poisson process in IRd, 31 in Wold process, 93 renewal process, 75 Regenerative homing set for Markov stationary, 320 chain, 93 nonstationary thinning, 329 Regenerative measure, renewal-type thinning by marks with regularly vary• example, 320, 354 ing tail, 294 Regenerative process, convergence to Random translation, equilibrium, 490 as cluster operation, 241 coupling argument, 491 effect on Bartlett spectrum, 416 splitting technique, 490 of Poisson process in IRd, 32 Regular conditional distribution, 641 point process invariant under, 306 Regular conditional probabilities, 604 Random translations, iterated, 300 existence, 166, 605 weak convergence, to Poisson process, Regular measure, 610 303 Regular point process on IR, 496 to mixed Poisson process, 305 conditional probability densities and Random walk, 67 survivor functions, 501 as point process, 123, 128, 207 conditional intensity function, 503 for symmetric stable distribution, 68 Regular representation, recurrent, 67 cluster process, 324 transience for stability of cluster stationary infinitely divisible point mechanism, 311 process, 322 transient, 67 Regular infinitely divisible process, see Random walk cluster process, 246; see Infinitely divisible also Bartlett-Lewis process Relative compactness for weak conver• Random walk point process, 67 gence, 619 Rarefaction of point process, see Random of Radon measures, 631 thinning Relative conditional intensity, see Rela- Rational spectral density, point process tive second-order intensity with, 414, 433, 439 Relative entropy, 564 Bartlett spectrum, 414 Reliability theory, 6 canonical factorization, 433 Renewal density, 6, 68, 84 linear prediction, 439 conditions for convergence, 88 intervals, exponentially distributed, convergence, 84 102 two-dimensional renewal process, 68 Recurrence time r.v.s, 71; see also Back• Renewal equation, 6, 65 ward and forward d-dimensional generalization, 67 joint distribution in renewal process, 74 for renewal density, 84 Reduced Campbell measure, 578; see also general, 65 Palm measure solution via martingale method, 70 convergence, 468 traditional form, 84 Reduced covariance measure, 357 unique equation with linear solution, Fourier transform, 411 66 mixing condition for convergence to Wold process analogue, 91 Lebesgue measure, 488 Renewal function, 64 Reduced factorial covariance measure and absolutely continuous, 84 density, 364 as solution of renewal equation, 65 Reduced moment and cumulant measures, bounds on, 89 372 density, 84 asymptotic behaviour, 372 one-one correspondence with renewal relation to Palm distribution, 464 process, 67 renewal process, 372 perturbation expansion for, 93 shift reflection property, 357 rate of convergence, 88 stationary process, 357 Renewal measure, 64 698 Subject Index Renewal process, Repeated conditioning for conditional as reduction from fermion process, 140 expectations, 640 as random walk with nonnegative Right-continuous histories, 515 steps, 207 Ripley's K-function, 383 backward recurrence time, 73 Ryll-Nardzewski's theorem, 475 joint distribution with forward recur- as extension of Palm-Khinchin equa• rence time, 74 tions, 477 Bartlett spectrum, 412 central limit theorem, 69 characterized by hazard measure, 103 Sandwich argument, continuous time characterized by renewal function, 65 result from, 331 compensator, 523 Scale invariant Poisson process and ran• complete intensity function, 560 dom measure, 325 conditional intensity function, 512 Schwartz distribution, random, 184 conditional p.g. fl., 233 Second moment measures, 410 convergence to equilibrium, 490 Bartlett spectrum, 411 current lifetime r.v., 73 characterization of simple point definition of counting measure, 64 process, 224 deterministic, 74 spectral representaion, 427 entropy, as approximation to stationary Second-order intensity, 365 point process, 572 relative, 365 forward recurrence time, 66 Neyman-Scott process, 373 higher moments, 69 Second-order theory for stationary gener• in discrete time, compensator, 526 alized process, 184 infinitely divisible characterization, 79 Self-approximating ring, 613 Laplace-Stieltjes transform, of kth fac- existence, 614 torial moment, 70 in extension of random measure, 159 ofp.g.f., 70 Self-correcting process, asymptotically lifetime distribution, 64 normal estimators, 548 likelihood function, 512 Self-exciting process, see Hawkes process mixing properties, 352 Self-similar random measure, 326 model for inhibition, 373 completely random case, 326, 330 moment densities, 70, 136 Separable space, 595 orderly, 64 Shear on cylinder, 386 characterization of Poisson process, in invariance of line process, 386, 390 74 Shift transformation, predicted time to next event, 107 on functions, 344 reduced moment measures, 372 S. on elements of Jfi9;, %9;, 317 stationary, 71, 320, 322 s. on measures on riJ( Jfi9;), 317 Palm-Khinchin equations, 52, 477 T. on points of fRd, 317 superposition, 76 Short range correlation property, 353 thinning, 75 Shot noise process, 188, 196, 263 two-dimensional, 68 as Neyman-Scott cluster process, 264 Renewal theorem, 80; see also Black• a-group, 325, 329 well's renewal theorem, elementary a-finite measure, 627 renewal theorem Signed measure, 596 absolutely continuous renewal function, Jordan-Hahn decomposition, 598 84 Simple counting measure, 197 convergence to equilibrium of point Simple function, 599 process, 488 Simple point process, 28, 44, 197; see Renewal theory, 1 , 63 also Orderliness Renyi's Poisson process characterizations, absolute continuity of compensator, 521 30, 31 coincides with support process, 209 Monch's extension, 216 example of nonorderly, 49 Subject Index 699 moment measure characterization, 135, Stationary infinitely divisible point 224 process, 322; see also Infinitely ordinary when boundedly finite, 214 divisible point process Simple stationarity, see Crude stationar• Stationary line process ity in plane, 385 Simulation of point process, associated line measure, 388 nonstationary Poisson, 23 first moment measure factorization, with bounded conditional intensity, 506 387 Wold, with gamma distributed inter- process of intersections, 397 vals, 102 second moment properties, 391 Singular infinitely divisible point process, in IR 3, 397 322 Stationary point process-basic proper• infinite KLM measure example, 330 ties (see also individual entries) weakly/strongly classification, 352 counter examples Singularity of measures, 601 cluster process with nonstationary Skorohod metric, 278 constituents, 329 Smoothing function, 401, 430 first- but not second-order, 371 Soft core point process model, 125 nonergodic, 60 Spectral density, one dimensional distributions, 329 perturbation expansion for, 93 second- but not first-order, 371 rational, point process with, 414, 433, definition and fidi distributions, 42, 439 318 Spectral measure and theory, 399 kth order, 355 generalized random functions, 400, 418 ergodicity, 58, 341 mean square continuous process, 401 equivalent to ergodic Palm distribu• multivariate random measure, 441 tion, 485, 493 point process or random measure, see invariant a-algebra, 330 Bartlett spectrum metric transitivity, 330 random measure in IR 2, 409, 450 ergodicity equivalence, 342 random measure on circle, 445 relation to stationary interval process, random signed measure, 415 475 wide sense random measure, 415, 420, zero/infinity sample path property, 322 421, 426, 429 for marked point process, 385 Spectral representation, 418 Stationary point process-classes or types limitation on usefulness, 424 (see also individual entries) random processes, 418 cluster, 329 second moment measure, 427 Cox, 329 Splitting technique, Markov chain, 93 deterministic, 329 Spread out probability distribution, 85 infinitely divisible, 322, 330, use in Wold process, 97 marked, 375 Stable cluster mechanism for iterated Neyman-Scott, 324 clustering, 310 planar, 378 Stable convergence, 644 renewal, 320 convergence of conditional distribu• as delayed renewal process, 71 tions, 496 recurrence times, 71 ar:mixing, 646 simple, Palm measure, 4 74 selection theorem for, 647 Stationary point process-functions or Stable process, in characterization of sta• properties of (see also individual tionary self-similar random meas• entries) ure, 328 a.s. events, Palm measure of, 485 State space of random measure, 154 avoidance function, 319, 328 Stationarity, 42, 316 entropy and entropy rate, 566, 569 Stationary generalized process, second• ergodic limits from Palm distributions, order theory, 184 485 700 Subject Index Stationary point process-functions or Palm distribution, 468 properties (cont.) u.a.n. triangular array, 281 expectation function, 58 Support counting measure, 198 hazard function of interval distribution, Support point process, 199, 208 56 characterized by avoidance function, intensity, 44 217 intervals, 42, 475; see also Palm distri- Supremum metric for d.f.s, 280 butions and measure Survivor function, 2 mean density, 43 Poisson cluster process, 244 support of Campbell measure, 469 Poisson process, 19 variance function, 372, 373 of conditional probability densities for Stationary random measure, 318 regular point process, 501 conditions for, 318 Symmetric difference of sets, 592 Palm theory, 460 Symmetric stable distribution, random Statistical ergodic theorem, 330 walk with, 68 for d-dimensional shifts, 333 System identification in point process, Stirling numbers, 112 443 recurrence relations for, 120 relation to kth moment measures, 139 Tail event, 343 Stochastic continuity sets, 270 Tail a-algebra !T<», 343 form an algebra, 271' 280 contains invariant a-algebra, 353 Stopping time, 650 infinite divisibility, 353 history prior to, 654 short range correlation, 353 of boundedly finite point process, 515 Taylor series expansion, Stress release model, see Self-correcting characteristic functional of random process measure, 190 Strict stationarity, 318 cumulant generating function, 114 Strong aperiodic recurrence of Markov p.g.f., 120 chain, 94 p.g.fl., 222 Strong convergence, 615; see also Con• Telegraph signal, 528, 532 vergence in variation norm Tempered measure, 409 of recurrence times in Wold process, 99 8-stationary process, 385 of u.a.n. array, 289 Thinning, see Random thinning, Strongly singular process, 352, 474; see Tight measure, 611 also Infinitely divisible uniformly, 619 Sub- and superadditive function, 60 random linear functional, 185 Subadditive set function, defines Topological groups, 631 extended measure, 210, 213 characters of locally compact Abelian, Sub- and supermartingale, 653 637 Superposition, 14, 229, 281 Topology, 593 infinite divisibility, 255, metric, 594 of independent point processes, product, 602 Bartlett spectrum, 416 quotient, 633 first limit theorem, 14 weak hat, 628 Grigelionis' Poisson limit, 285 Total boundedness of c.s.m.s., 595 infinite, condition to be well-defined, Total variation of measure, 598 233 Totally finite measures as metric space, p.g.fl., 229 weak convergence and equivalent renewal processes, 76 topologies, 623 stationarity and characterizations, totally finite measures on c.s.m.s., 624 75 as metric space, 625 of random measures, 193, 281 c.s.m.s. properties, 627 generating functionals, 194 equivalent topologies, 623 Subject Index 701 Transformable measure, 401 stationary random measure, 372, 373 Translation bounded on circle, 445 measure, 401, 409, 429 on cylinder, 386 sequence, 409 on plane, 378 Translation, see Random translation on cone, 385 Triangular array, see U.a.n. array Variance/mean ratio, 365; see also Over• Trigger process, 263 and underdispersion Trivial a-algebra(= metric transitivity), asymptotics, 372 342 negative binomial distribution, 11 Two-point Poisson cluster process, Variation metric, 296 Bartlett spectrum and canonical factori• Variation norm, 598 zation, 433 properties, 280 linear prediction, 440 convergence in, 272, 615 Vehicular traffic process, 378 Velocities, particles with random, 307, 378 U.a.n. (uniformly asymptotically negli• Voronoi polygon in IRd, 480 gible) array, 281 convergence, p.g.fl. conditions, 289 Grigelionis' Poisson limit, 285 Waiting-time paradox, multivariate process, 287 Poisson process, 20 random measures, 288 length-biased sampling explanation of, strong convergence, 289 42,55 weak convergence, 283 Weak asymptotic uniformity of sequence Unbounded measure, Fourier transform of measures, 307, 314 theory, 399 Weak convergence of measures, 615 Underdispersion, 367, 373; see also of random measures, 271 Overdispersion convergence of associated cumulative Uniform integrability, 643 processes, 278 equivalent to L convergence, 644 1 convergence of fidi distributions, 272 Uniform random measure, 156 equivalent conditions for, 276 Uniformly asymptotically negligible, see point processes with simple point U.a.n. process limit, 277 Uniformly tight measures, 619 relative compactness, 619 random measures, 273 weak* convergence, 615 Unordered sets of points defining point Weak hat topology, 628 process, 121, 126 relative compactness, 630 Updating formulae in prediction, 539, Weakly singular process, see Infinitely 548 divisible Urysohn's theorem, 595 Weibull distribution, 3, 7 Weighted average version of ergodic the• orem, 339 Vacuity function, 215; see also Avoidance Wide sense random measure, 415, 421, function 424 Vague convergence of measures, 615 Bartlett spectrum, 426, 429 Radon measures, 631 spectral measure, 415, 426, 429 Variance function, 58, 365, 373 Wiener's homogeneous chaos, 161 absolutely continuous random measure, Wold decomposition, 431 372 random measure, 430 analytic properties, 373 Wold process, 89 cluster process, 373 asymptotic properties, 95 Fourier transform, 411 Blackwell's renewal theorem analogue, Poisson process, 373 95, 98 renewal process, 77 characterized by hazard measure, 103 702 Subject Index Wold process (cont.) interval structure, 90 complete intensity function, 560 regenerative methods for, 93, 103 conditional intensity and likelihood, renewal equation analogue, 91 504 simulation, correlated gamma dis- conditionally exponentially distributed tributed intervals, 102 intervals, 102, 107 stationary, 90 convergence to equilibrium, 490 Palm-Khinchin equations, 478 discrete, 91, 101 exponential intervals, 89, 93, 102 hazard function, 512 Zero/infinity property, higher-order, 103 stationary random measure, 322 infinite intensity example, 99 marked point process, 385