APPENDIX 1 Existence of random processes with given finite-dimensional distributions Let X = X(t,w) be a stochastic process in the nota­ tion of Chapter I, section 1, with state-space Sand para- meter set T. Let ~ denote the mapping from the t l' ... , tn probability space n into the n-tuple (X(tl,w), ... , X(tn,w». Then the finite-dimensional distributions of the process X are the probability measures defined by P (C) = p(~-l C) (1 ) tl.···.tn t l •··• .tn where C is any measurable set in Sn; this is the same as (1) on page 2 and makes sense since ~ must be t l •·· .• tn an ~-measurable mapping. Such a measure P is t l •· .. ,tn defined for each finite. ordered subset of T. The measures defined by (1) must satisfy certain con­ sistency conditions. Let w denote any permutation of the integers I.2 •...• n and let f be that one-to-one mapping 'If of Sn into itself defined by n. f or any n-tup 1 e ( xI ••.. ,xn ) E S It is clear that t't t (w) = f • fJ t t (w), 'If 1 •••• , 'lfn 'If 1····' n and from this it follows that the distributions defined in (1) must satisfy (2) Appendix 1 251 This is the first consistency condition. Now define a n+m,n to be the projection mapping from Sn+m to Sn which sends (x , ... ,x , ... ,x ) into 1 n n+m (xp,,,,x ); then 0-1 C consists of all those points in n n+m,n Sn+m whose first n coordinates determine a point of the set C l'n 8n. For any or d ere d set 0 f n+m e 1 ements 0 f T we then must have and so from (1) it follows that (3) n for all measurable C C 8 and all n and m > O. This is the second condition. The finite-dimensional distributions of any stochastic process on T with state-space 8 must automatically satisfy (2) and (3). The problem now is to prove a converse. That is, sup­ pose we are given any set T, any measurable space (S,S/), and a family of measures {p } on the measurable t l ,· ., , tn sets of the products 8n which satisfy both conditions (2) and (3). Does there exist a random process X = X(t,w) on some probability space which has parameter set T, state­ space 8 and whose finite-dimensional distributions are the given family? The answer, in this generality, is "no." but when some assumptions are made about (8,5') a positive re­ sult can be proved. We will sketch the proof with S = Rl and then discuss briefly how far it can be generalized. Theorem. (Kolmogorov). Given any ~ T and any family of measures 252 APPENDIX 1 ~ Euclidean spaces Rn which satisfy (2) and (3), there exists a probability space (n, .5',P) ~.! real random pro­ cess {Xt : t E T} defined ~ it such that condition (1) holds; i.e., the measures {P } are the finite-dimen- -- - tl, .. ·,tn ----..........;:.:-..:. sional probability distributions of {Xt }. Sketch of Proof. We use a direct product to define our measurable space (n,.5'), although of course P will not be a product measure except in the ·special case when the ran­ dom variables are to be independent. Accordingly we put TR n = II Rt (R,Rt is the real line); (4) tET a typical element of n will be a function w('): T 0+- RI. The field .5' will be the 0- field generated by all sets of the form E' = {w En: (w(t ),·· • ,w(t » E e} = f1 (e), (5) l n t 1 , ... , tn n where e is some Borel set in R and (tl,···,tn) is any finite ordered subset of T; these are called (Borel) cylin- der sets. The random variables of the process we are trying to construct will be the coordinate functions defined by X(t,w) = wet), t E T, (6) and so equation (1) must be used to define the measure P on the class of cylinder sets (5). There are two obstacles to be overcome. First, we must show that P is unambiguously defined on the cylinder sets. That is, if two· different subsets of T and/or Borel sets e can be used in (5) to define the same cyiinder set E it must be verified that the proposed finite-dimensional Appendix 1 253 distributions. used in (1). assign the same measure to E regardless of which representation is chosen. It is precisely here that the consistency conditions (2) and (3) come in; we omit the details. Second. the measure P which is now defined for cylin­ der sets must be extended to a true countably-additive prob­ ability measure on the a-field ~ According to the basic "extension theorem" which is fundamental in measure-theory. this ~ be done. and uniquely. provided that whenever En is a decreasing sequence of cylinder sets with nE n n we have lim (7) n + 00 To establish this "continuity condition." suppose that (7) does not hold so that P(E) > d > 0; we will show that then n the intersection of the En can't be empty. Each cylinder set En is defined by (5). using a cer­ tain ordered finite subset Tn of T and a "base" Cn eRn. It is easy to see there is no loss of generality in assuming that the sets Tn are increasing. It is also possible. though less trivial. to assume that each Cn is compact. The idea here is that if one of them is not compact. because the measure P (.) must be regular on Rn that set t l •··· .tn C can be replaced by a compact subset C' whose measure n n differs arbitrarily little from P (C ) = peE ). t l •· ... t n n n The corresponding new cylinder sets E' can be made to n satisfy the same conditions as the original E. and in par­ n ticular can be chosen so that peE') > d/2 for all n. n - Now it is possible to show that n E + .; clearly n it is enough to show n E~ +~. For each n. choose a point 254 APPENDIX I Wn E E'. By the definition of E'I we have ~T (wI) E C" n I I' since W E E' C E' we also have ~T (wn ) E Ci· Since C' n n I I I is compact, there is a subsequence {wn ' } such that lim w, (t) exists for each t E TI and these limits are n' .... co n the coordinates of a point in Ci. Next, by a similar argu- ment it is possible to choose a sub-subsequence {wn,,} such that the sequences {wn" (t)} have limits for each t E T2 and the limits define a point in CZ' Continue this process for all n, and then use Cantor's "diagonal argument." The result is a subsequence {wn*} such that lim n* .... 00 exists for each t E UTn; moreover, if (tl ,t2, ... ,tk) = Tn then (w(tl ), .•. ,w(tk)) E C~. Finally, define wet) = 0 (or any other fixed value) when t Et: UTn . The function w now belongs to every set E', and the continuity condition is n proved. The extension theorem then gives us the desired measure on ~ and Kolmogorov's theorem is established. As noted, the theorem is not true when the real line is replaced by an arbitrary measurable space. The proof above, however, still works if a locally-compact metric space is used. The critical point is that any finite Borel measure on such a space must be regular, so that measurable sets can be approximated from within by compact sets. This gen­ eralization suffices for the processes studied in this book. APPENDIX 2 Review of conditional probability Let (n, ~,P) be a probability space, ~' a a­ additive subalgebra of ~ and X E Ll an integrable random variable. (The Ll space may be real or complex.) We call YELl a version of the conditional expectation of X with respect to gr', and wri te Y = E (X! Y'), (1) provided that (i) Y is measurable with respect to .$I' and (ii) for every A E Y' we have f YdP = f XdP. A A When X is the characteristic function ~B of a set B E ~ Y is called the conditional probability of B and written Y PCB! gr'). (2) If gr' = IB{X : a E Q} is the a-algebra generated by a set a of random variables {X a , a E Q}, we can write Y=E(XIXa:aEQ) or Y=P(B!Xa:aEQ) instead of (1) or (2) and speak of the conditional expecta- tion (or probability) "given" the random variables {X }. a Some of the properties of conditional expectation (enough for the purposes of this book) are listed below: Proposition. Let X E Ll and.$l' C 9 be given as above. 256 APPENDIX 2 (a) and Y are both versions of E(xl~l) 2 then YI = Y2 a.s. (b) If X is measurable with respect to ~', then E(XIJFt) = X; if JF' ~ {.,O}, then E(XljTl) = E(X) • (c) Conditional expectation is linear in X. (d) If X ~ 0, E(xljPl) ~ 0 (a.s.). Consequently, IE(XI~I)I ~E(IXIIJFI) a.s. (e) If Xn + X ELI' and if either {Xn} increases or {Xn} is dominated, then E(XnljTl) + E(XljTl) a.s. (f) If X £ Ll and if X is independent of every set in~~, tl E(x~~) = E(X) a.s.