IJST (2012) A4: 461-467 Iranian Journal of Science & Technology http://www.shirazu.ac.ir/en
Conditional expectation of weak random elements
Z. Shishebor1*, A. R. Soltani2, M. Sharifitabar3 and Z. Sajjadnia4
1Department of Statistics, Shiraz University, Shiraz, Iran 2Department of Statistics, Shiraz University (and Kuwait University) Shiraz, P.O. Box 5969 Safat 13060, Iran 3School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran 4Department of Statistics, Shiraz University, Shiraz, Iran E-mails: [email protected], [email protected], [email protected], & [email protected]
Abstract
We prove that the limit of a sequence of Pettis integrable bounded scalarly measurable weak random elements, of finite weak norm, with values in the dual of a non-separable Banach space is Pettis integrable. Then we provide basic properties for the Pettis conditional expectation, and prove that it is continuous. Calculus of Pettis conditional expectations in general is very different from the calculus of Bochner conditional expectations due to the lack of strong measurability and separability. In two examples, we derive the Pettis conditional expectations.
Keywords: Pettis integral; Pettis conditional expectation; non-separable Banach spaces; weak p-th order random elements
1. Introduction not have conditional expectation with respect to the -field of Lebesgue measurable sets. Riddle and Recently, studies of infinite dimensional processes Saab [7] prove sufficient conditions for the have increased dramatically due to the progress of existence of Pettis conditional expectation for technologies which allow us to store more and more scalarly measurable bounded random elements for information while modern instruments are able to all -fields inside the -field of the underlying collect data much more effectively due to their probability space. In this paper, we go for the increasingly sophisticated design [1]. Although the Riddle and Saab settings, and establish basic strong second order processes are well developed ingredients for the calculus of the Pettis conditional and widely known [2], the weak second order expectation of weak first-order scalarly measurable processes are not rich enough in theory. This, random elements with values in the dual space of a strongly motivates us to define the conditional non-separable Banach space. The upshot is the expectation of weak second order random processes continuity property, established here. This article is which play a crucial role in the study of different organized as follows. In Section 2 we provide subjects such as martingale theory. notations and preliminaries. Certain new results are A few authors tried to develop the theory of Pettis also provided, Lemma 2.2 and Lemma 2.3. Section conditional expectation of p-th order random 3 is devoted to the main results, basic properties of elements. Brooks [3] gives a formal representation the conditional expectation are given in Theorem of the conditional expectation of strong measurable 3.1, and the continuity is given in Theorem 3.2. and Pettis integrable random elements for a given -field. Uhl [4] provides sufficient conditions for the existence of the Pettis conditional expectation. 2. Preliminaries There are also counter examples which show that Let C be the space of complex numbers. Suppose the Pettis conditional expectations do not exist in general, (see Raybakov [5]). Also, Heinich [6] that X and Y be complex Banach spaces, B(X ) provides an example of a Banach space valued stands for the Borel -field: the smallest -field Pettis integrable function on [0,1]2 which does generated by open subsets of X . The notation x , x is used to denote x (x) when x X *Corresponding author Received: 26 December 2011 / Accepted: 18 February 2012 and x X , where X is the dual space of X . Let (,F,) stand for a probability space.
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A Y -valued function (random element) where fn d is defined in the obvious way and : Y is called strongly measurable when it E (B) f d is called Bochner integral of f is F/B (Y) measurable. A random element from E into Y is called scalarly measurable if the with respect to . complex-valued random variable y , is Let Ł p (Y,) , then E := (B) d B measurable, i.e. F/B(C) is measurable for every is defined in the sense of Bochner integral which 1 y Y . Let us introduce some classical function defines bounded linear transformation of Ł (Y,) spaces which will be used throughout this paper. into Y . Also, the conditional expectation of p • Ł (Y,) stands for the space of all strongly given a -field Γ in F is defined as being a 1 measurable random elements on Y equipped random element in Ł (Y,), denoted by with the norm, EB [ | ], which is Γ -measurable and satisfies p 1/p the condition that
p = (E Y ) , (B) d = (B) E [ | G]d, for all A . A A B which are called strong random elements of order p . From now on, we use the term "random element" for scalarly measurable random elements only. • Lp (Y, ) 1 p < stands for the space of w scalarly measurable random elements in Y for Definition 2.2. A random element : Y is which, called Pettis integrable with respect to , if
w 1 p 1/p (i) Lw(Y,) , p = sup (E | y , | ) , * y 1 (ii) For every E F , there exists an element E in Y such that, is finite. Such a random element is called weak y, = y,d, for every y Y . (1) scalarly measurable random element of order p . E E
Evidently, the weak p -th order property is weaker The element E is called the Pettis integral of than the strong one. over E with respect to the measure and it is • Let L (X ,) denote the space of scalarly w denoted by (P) d . In particular, E E P measurable random elements in X equipped stands for . with the norm, w If Y is a reflexive Banach space, then every = sup | , x |, x X, X . esssup separably-valued random element of a weak order 1 x one is Pettis integrable, [8]. • L () stands for the space of all mappings from Y into C that are bounded a.e. Lemma 2.1. Let and be two random elements in Y . Then, Definition 2.1. A -measurable function (i) is a random element in Y . f : X is called Bochner integrable if there (ii) If A : Y Y is a bounded linear operator, exists a sequence of simple functions { fn } such then A is a random element in Y . that (iii) Let {n } be a sequence of random elements in lim f f = 0. Y such that a.e. (i.e., n n n y ,n 0, a.e. for every y Y ), In this case (B) f d is defined for each E then is a random element in Y . E F by (B) f d = lim f d Proof: The proof is straightforward. E n E n
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Definition 2.3. A sequence {yn } in a Banach 1 space Y is called weakly Cauchy if for any positive Lemma 2.2. Let n in Lw(X ,) , and number and every y Y , there exists a {n } be a sequence of Pettis integrable random positive integer N , depending on and y , such elements. Then is a Pettis integrable random that for all m,n > N , we have element in X . | y , yn ym |< . Proof: It is enough to show that for all A F , there exists X such that Definition 2.4. A subset K of the Banach space Y A is called weakly precompact if each sequence in K x ,d = x , . Since in A A n has a weakly Cauchy subsequence. 1 Lw(X ,) , Definition 2.5. Pettis integrable random elements and in Y , are called weakly equivalent if sup | x, |d 0, A n y , = y , a.e.,for all y Y , x 1 w = a.e . and since x , d = x , and A n nA Definition 2.6. Let be a Pettis integrable random sup | x, |d 0 , so, n m element in Y and let Γ be a -field in F . A Pettis x 1 integrable random element EP [ | ] in Y is said to be Pettis conditional expectation of with sup | x, | 0. nA mA respect to Γ if, x 1 (i) EP [ | ] is scalarly Γ-measurable and Pettis integrable, Therefore, { } is a weak Cauchy sequence in (ii) (P) d = (P) E [ | ]d, for nA G G P every G . X and converges weakly to some element Definition (2.6) agrees with the one given in [4] if X and x , = x ,d . A A A the random element is strongly measurable. From now on, we assume Y = X , i.e. the Lemma 2.3. Let be a random element in the desired random element takes its values in the dual space of a non-separable Banach space. This Banach space X for which EP [ | ] exists, enables us to use certain weak* properties. We note then that this assumption is satisfied whenever Y is a x,E [ | ] = E[x, | ] x X . Hilbert space or Y = . P The following theorem is given by L. H. Riddle Proof: and E. Saab [7]. It gives sufficient conditions for Since (P) E [ | ]d = (P) d for bounded Pettis integrable random elements to have G P G Pettis conditional expectation. all G , then
Theorem 2.1. Let : (,F,) X be a xx,[|]d=E ,d GGP bounded Pettis integrable random element. If the set =[,|]d,Ex forallG . {,x : x 1} is weakly precompact in G
L () , then has Pettis conditional expectation with respect to all sub- -fields in F . Since x ,EP [ | ] and E[x , | ] To establish the main properties of Pettis are -measurable functions, we conclude that conditional expectation, we first prove the x ,EP [ | ] = E[x , | ], a.e. following result.
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{,x : x 1} are weakly precompact sets in Corollary 2.1. Let be a random element in the L () . Then: Banach space X for which EP [ | ] exists, w (i) If = , a.e., then , ae.. then c EP [ | ]=c (ii) If k is a scalar, then EP [ | ],x = E[,x | ] for all x X. w
EP [k | ]=kEP [ | ] EP [ | ] , ae.. Proof: Apply Lemma (2.3) and the fact that X can w (iii) If = {,} then E [ | ]=E , ae.. be embeded in X . P P w
(iv) E [ | F]= , ae.. Lemma 2.4. Let X be a Banach space, then P w {x X : x 1} is weakly precompact if and (v) If 1 2 then EP [EP [ | 2 ] | 1]=EP [ | 1] , ae.. only if {x X : x M } is weakly precompact. (vi) If A is a bounded linear operator on X , then Proof: The result follows from the fact that w EP [A | ]= AEP [ | ] , a.e.
x x {x ,x : x M } = {x , : 1}, Proof: (i), (ii) and (iii)are immediate from the M M linearity and basic properties of Pettis integrals, and by Definition (2.6). Lemma 2.5. Let be a bounded random element For (iv) we have
in the Banach space X . If {,x : x 1} is x,E [ | F]d = x,d, for all G F. G P G weakly precompact in L () , then EP [ | ] is Since x ,EP [ | F] and x , are bounded and {EP [ | ],x : x 1} is weakly measurable F , we obtain precompact in L () . x ,EP [ | F] = x ,, Proof: According to the assumptions of the Lemma w (2.5) and Theorem (2.1), EP [ | ] exists. Since which means EP [ | F]= , a.e. {,x : x 1} is weakly precompact set, so for For (v) we note that scalar measurability and each sequence {, xk } there is a subsequence weak integrability of EP [ | 1] are immediately {, x } which is weakly Cauchy in L () . followed by Definition (2.6), k(i) Also, by Lemma (2.3) and the fact that X is x ,[[|]|]dEE G PP 2 1 embedded in X [9], we have E [ | ],x x = E[,x x | ], =,[|]d,xforallGE . P k(i) k( j) k(i) k( j) G P2 1 and so E[,x x | ] 0 , since k(i) k( j) Since 1 2, we obtain that ,x x 0 in L () . k(i) k( j) xx,EE [ [| ]|]d= , E [| ]d GGPP 2 1 P 2 =,d,x 3. Main Results G =,[|]d,x E We first derive some important properties of Pettis G P1 conditional expectation, given in the following for every G 1 . theorem. w Theorem 3.1. Suppose that and are bounded So EP [EP [ | 2 ] | G1]=EP [ | 1] , a.e. scalarly measurable and Pettis integrable random For (vi), the existence of EP [A | ] is elements such that {,x : x 1} and guaranteed by Lemma (2.4). By Definition (2.6):
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xA,[EP |]d= xA , d Now we need to show that {, xk }i is also GG ii =,dAx weakly Cauchy. For simplicity we let xk = yi . G ii * * =,[|]dAx E Let > 0 and f L () be given, there G P w =,[|]d,xA E exists n such that < /(4 f ) , so G P n forevery G ; | f (n , yi y j ) |< , for every i, j. hence 2 Since { , y } is weakly Cauchy for this choice w n i i EP [A | ]= AEP [ | ], a.e. of n , there exists a scalar M such that for every
The following theorem is the main result of this i, j > M , | f (n, yi y j ) |< . Therefore, article. It gives the continuity of the Pettis 2 conditional expectation under the Riddle and Saab |(,fyyfij )||( n , yy ij )| assumptions given in [7]. |(,fyyni j ,)|<, ijM ,>; Theorem 3.2. Let {n } be a sequence of bounded This leads us to the weak precompactness of random elements of weak order one in the Banach {,x, x 1} . Hence EP [ | ] exists. space X . Let {n,x, x 1} be a weakly To prove the convergence, using Lemma (2.3) we have precompact set in L () and n in 1 w* w L (X ,) L (X ,), i.e, 0 w w n n 1 sup esssup | x ,EP [n | ] EP [ | ] | as n , where is a bounded random x 1
element on X , then EP [ | ] exists and = sup esssup | x ,EP [n | ] | x 1 EP [n | ] EP [ | ] in Lw(X ,) .
= sup esssup | E[x ,n | ] | Proof: To prove the existence, since n and are x 1 1 in Lw(X ,) , it will be enough to show that sup esssup | x ,n | {,x, x 1} is a weakly precompact set in x 1
L () . Let {, xi }i be an arbitrary 0, asn , subsequence of {,x, x 1} . Since for every The last two assertions follow from the fact that , { ,x, x 1} is weakly precompact, 1 m m n in Lw(X ,) Lw(X ,) , and the contraction of the conditional expectation [10]. then for x1,x2,... in X such that xi 1, the Hence EP [n | ] EP [ | ] in Lw (X ,) . sequence {m, xi }i has a subsequence Let us conclude this section by presenting two { , x } which is weakly Cauchy, m kmi i examples which are not strongly measurable [11], m = 1,2,... . We prove by induction that there pages 44-45 and [12]. We obtain the Pettis conditional expectation of the corresponding exist a sequence {x } for which k(m)i i random elements.
{ , x } is weakly Cauchy. Thus by the m kmi i Example 1. Let ([0,1], B([0,1]),λ be a probability diagonalization method { , x } is weakly m kii i space where is the Lebesgue measure on [0,1] Cauchy for each m . and B([0,1]) is the Borel -field on [0,1] . Also,
let {et ,t [0,1]} be an orthonormal basis for the
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non-separable Hilbert space l2 [0,1], (the space of 1 E[ A | ] = { A (t)dt}G . n Gm n m all complex valued functions on [0,1] which m (Gm ) disappear everywhere except at the most countable The measurability of ,EP [ f (t) | ] is points of [0,1] , and the sequence of values at obvious because of the structure of Γ. For the those points is square summable.) second condition of the Definition (2.6), it is To find the conditional expectation of a random enough to check the equality for one Gi ; element f : [0,1] l2 [0,1] defined by f (t) = et , we use Riesz Representation Theorem ,(Ett [ | ]( )) d G Ann to obtain f ,x = 0 , a.e., for every i 1 x l2 [0,1] . Thus f is a random element in =,(An ()d)dtt t GGn ii()Gi l2 [0,1]. It is obvious that its conditional expectation with respect to any sub- -field of 1 =(GAGinin ) ,(( )) B([0,1]) is zero, which is Γ -measurable and ()Gi satisfies the integral equation of conditional expectation. =,((AGnin )).
Also, we need to show that Example 2. Suppose that {An } is a sequence of subintervals of [0,1] , which has the following ,( (tt )) d G Ann properties: i =,(())d tt (i) A1 = [0,1] , 1 An Gi n (ii) each An is a nonempty subinterval of [0,1] , 2 ,(An (tt )) d Gi n (iii) lim(An ) = 0 , where is the Lebesgue measure, =,((AGnin )).
(iv) An = A2n A2n1 for all n , Since (An ) 0 as n , so (v) Am Aj = for each pair (m, j) with (An Gi ) 0 as n , hence i i1 2 m < j 2 1, for some i . ,((A G )) = 0. Therefore, 2 n i n Let f :[0,1] l l1 be defined by ,( A (t))n dt = 1,( A (t))n dt f (t) = ( (t)) ,t [0,1] and let l . By Gi n Gi n An n = 1,( ( A (t)dt)n Yosida-Hewitt Theorem, there exist a unique 1 Gi n and 2 in l such that =1 2 , where 1 is = 1,((An Gi ))n countably additive part of and 2 is the purely = ,((An Gi ))n . finitely additive part of . It is shown in [10] that Thus (E[ | ]) is a version of f (t) is a non-measurable scalarly measurable An n function. Now we find the Pettis conditional EP [ f (t) | ] . expectation of f with respect to the -field Γ, generated by a countable partition {G1,G2,} , of References [0,1] . Indeed (E[ | ]) is a version of An n [1] Dabo-Niang, S. & Ferraty, F. (2008). Functional and Operatorial Statistics. Physica-Verlag, Heidelberg. Pettis conditional expectation of f , where [2] Bosq, D. (2000). Linear processes in function spaces: E[ | ] is the usual conditional expectation of theory and applications. Vol. 149, Lecture notes in An statistics, New York. Springer-Verlage. random variable with respect to Γ. To verify [3] Brooks, J. K. (1969). Representations of weak and An strong integrals in Banach spaces. Proc. Nat. Acad. sci. this, it is easy to show that USA, (2), (266-270).
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[4] Uhl, JR. J. J. (1972). Martingales of strongly measurable Pettis integrable functions. Trans. Amer. Math. Soc., 167, 369-378. [5] Raybakov, V. I. (1971). Conditional mathematical expectation for functions that are integrable in the sense of Pettis. Mat. Zametki, 10, 565-570. [6] Heinich, M. H. (1973). Espérance conditionelle pour les fonctions vectorielles, C. R. Acad. Scie. Paris Sér. A , 276, 935-938. [7] Riddle, L. H. & Saab, E. (1985). On functions that are universally Pettis integrable. Illinois J. Math., 29(3), 509-531. [8] Vakhania, N. N., Tarieladze, V. I. & Chobanyan, S. A. (1987). Probability distributions on Banach spaces. Holland, D. Reidel Publishing Company. [9] Rudin, W. (1991). Functional analysis, second edition. McGraw-Hill. [10] Carothers, N. L. (2005). A short course on Banach space theory. Cambridge University Press. [11] Uhl, J. J. & Diestel, J. (1979). Vector measures, second edition. American Mathematical Society. [12] Musial, K. (2002). Pettis Integral, Handbook of Measure Theory, Chapter 12, Elsevier science.