Research Article | Open Access
Strong Law of Large Numbers of Pettis-Integrable Multifunctions
Using reversed martingale techniques, we prove the strong law of large numbres for independent Pettis-integrable multifunctions with convex weakly compact values in a Banach space. The Mosco convergence of reversed Pettis-integrable martingale of the form , where is a decreasing sequence of the sub -algebra of is provided.
The strong law of large numbers (SLLN) is used in a variety of fields including statistics, probability theory, and areas of economics and insurance. In recent years, SLLN has been extensively studied by several researchers. Let us mention Artstein and Hart , Castaing and Ezzaki , Etemadi , Ezzaki , Hess , and Hiai .
In the theory of integration in infinite-dimensional spaces, Pettis-integrability is a more general concept than that of Bochner-integrability. The purpose of this paper is to prove the SLLN for measurable and Pettis-integrable multifunctions by using the techniques of reversed martingale. The proof is based on the recent properties of Pettis-integrable multifunctions. See for example Akhiat et al. , Chowdhury , El Amri and Hess , Geitz et al. , Thobie and Satco , and Musial .
The paper is organized as follows.
2. Notations and Definitions
Throughout this paper, we assume that is a complete probability space and is a decreasing sequence of sub--algebra of , such that . is a separable Banach space with the dual space .
Let be a dense sequence in with respect to the Mackey topology , (resp., ), and the closed unit ball of (resp., ). Let (resp., ) be the family of all nonempty convex and closed (resp., convex weakly compact) subsets of .
Given in , the distance function and the support function associated with are defined by For any in , we get A -valued sequence is called Mosco convergent to a closed convex set if , where If Mosco converges to in , we write
A measurable function is Pettis-integrable, if is scalarly integrable
(i. e. is integrable), and, for each , there exists in , such that is called the Pettis-integral of over . We will denote by the space of all measurable and Pettis-integrable -valued function defined on . We consider the space provided with the following topologies.
(i) The topology of the usual Pettis norm is as follows:
(ii) The topology is induced by the duality Recall that a sequence in converges to in this topology, if, for each and for all , one has
This topology is known as the weak topology and is denoted by -.
A multifunction is said to be measurable, if for every open set of the subset is an element of . The Effros -field on is generated by the subsets , so is measurable if, for any , one has .
Two measurable multifunctions and are said to be equal scalarly almost surely, if the following equality holds: For any convex and weakly compact measurable multifunction , the measurability of is equivalent to that of its support functional . (See .)
The tribe trace of on is defined by the set and is denoted by .
A measurable function is said to be a selection of , if, for any , . We denote by the set of all measurable selection of . It is known that a convex and closed valued multifunction in separable Banach space is measurable if and has a Castaing representation (i.e., there exists a sequence of measurable selections of such that for all , .
The distribution of the measurable multifunction on the measurable space is defined by Two measurable multifunctions and are said to be independent, if the following equality holds: The measurable multifunction is scalarly integrable, if, for any , the real function is integrable.
We say that the measurable multifunction is Pettis-integrable, if it is scalarly integrable and, for each , there exists such that is called the Pettis-integral of over . Let be the set of all -measurable and Pettis-integrable selections of .
The multivalued Pettis-integral of a -valued multifunction is defined by and is convex and compact and, for each , we have We will denote by the set of all Pettis-integrable -valued multifunctions.
Given a sub -algebra and a -valued Pettis-integrable multifunction, the Pettis conditional expectation of with respect to is a -measurable -valued Pettis-integrable multifunction denoted by which satisfies The two following propositions (see ) give a sufficient condition of the existence of the conditional expectation for a Pettis integrable -valued function and for a -valued Pettis-integrable multifunction.
Proposition 1. Assume that is separable. Let be a sub -algebra of and let be a Pettis-integrable -valued function such that Then, there exists a unique -measurable Pettis-integrable -valued function denote by , which enjoys the following property; for every , one has
Proposition 2. Assume that is separable. Let be a sub--algebra of and let be a -valued Pettis-integrable multifunction such that Then, there exists a unique -measurable -valued Pettis-integrable multifunction, which enjoys the following property; for every , one has
Akhiat et al.  extended the previous theorem in a -valued Pettis-integrable multifunction.
We close this section by the following useful corollaries.
Corollary 3 (see ). Let be a decreasing sequence of a sub--algebra of and let ; set . Then,
Corollary 4. Let be a sub--algebra of and let be a -valued Pettis-integrable multifunction such that Then, for all , the following properties hold:(1) almost surely.(2) almost surely.
Proof. (1) Let and be a dense sequence in with respect to the Mackey topology , then so and hence We conclude that For any , we have , then, by Proposition 2, we have . Since is a Pettis-integrable multifunction, And hence by uniqueness of the conditional expectation of , we obtain
3. Strong Law of Large Numbers for Pettis-Integrable Multifunctions
Our first result is the following theorem.
Theorem 5. Let be a decreasing sequence of sub--algebras of and set . Let be a separable Banach space and such that . Then
Proof. Since , so, by Proposition 1, we have that exists and is in and provides a -measurable partition of such that (see ). Using Corollary 3, we obtain As , for every and for every , then On the other hand, Therefore,
Let us prove the following results which will be used after.
Proposition 6. Assume that is separable. Let X and Y be two -valued Pettis-integrable multifunctions and be a sub--algebra of such that and . Then
Proof. Let be a dense sequence in with respect to the Mackey topology . Since and are -valued Pettis-integrable multifunctions, then, for any in , there exist two sets and in , such that for all and
By Theorem 5.1.6 in , is -valued Pettis-integrable multifunction, then, for any in , there exists a set in such that On the other hand,Then
Therefore, On the other hand, By (31), then and therefore
Theorem 7. Let and be in ; let be a random variable with values in a measurable space such that and have the same distribution. If and Then
Proof. Since is Pettis-integrable with values in , . So by , there exists selection Pettis-integrable of , -measurable such that Let be a fixed element in , , and . Then, by Doob’s factorisation lemma, we can find a -measurable mapping from into (i.e., and a -measurable function from into (i.e., , which satisfy On the other hand, And By using the application And by the classical transfer theorem, we haveThen By combining relation (39), (43) and the fact that , we obtain Then In particular, if , for any , we have then and therefore And hence, by uniqueness of the conditional expectation of relative to ,
Before giving the principal result, we also need the following classical theorem (see p. 52 in ).
Theorem 8. Let and be two sub--algebras of , is a -algebra generated by and , and let an integrable real measurable function. If and are independent of , then
Now, we give the main result of this work.
Theorem 9. Assume that is separable. Let be a sequence of independent measurable multifunctions in
Let , , and assume that (i), and have the same distribution.(ii), . Then, we have the following assertions: (1), (2)(3)
Proof. The first equality follows from the Theorem 7.
Now let us prove the second equality. Let be a dense sequence in for the Mackey topology. Set , and .
Since and are independent of , and are independent of , then, by applying the Theorem 8, we have So, , then, by using Corollary 4, we obtain and since and are convex and weakly compact, then (i) Now, we show the last assertion.
Step 1. We claim that
We have , so, by Proposition 6 and the first and the second assertions of the theorem, we obtainThen Step 2. We show that
(ii) We begin by proving that
Since is Pettis-integrable, then there exists Pettis-integrable selection of such that On the other hand , then exists and is in . Let ; by Corollary 4, exists and a. s.
Using Theorem 5, we have a.s.
Since by  Then and by (56),
Then (iii) Now, we show that
Let be a dense sequence in for the Mackey topology; we have then, by Corollary 3, Hence, there exists a negligible set ; for all and for every , we have So, for all and for all , Let and , then there exists in such that , which implies
Then Consequently, , then This yields
Corollary 10. Under the same hypothesis of Theorem 9, we have
Proof. By the previous theorem, we need only to check that Since is convex and weakly compact. Now let be a dense sequence in for the Mackey topology; we have On the other hand, is a Pettis reversed martingale, so, for each positive integer , set, and . Hence, for all any , the multifunction is -measurable. Moreover, by the previous theorem, we have Then Since, for every fixed positive integer , we have
Then by (70) Since, for all fixed integer , the multifunction is -measurable and by (71) is -measurable and so is -measurable. Then by the independence of and the Kolmogorov’s Zero-One law (see ), we conclude that for all Since and are -valued multifunctions and (72) is true for all , then
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
- Z. Artstein and S. Hart, “Law of large numbers for random sets and allocation processes,” Mathematics of Operations Research, vol. 6, no. 4, pp. 485–492, 1981.
- C. Castaing and F. Ezzaki, “SLLN for convex random sets and random lower semicontinuous integrands,” Atti del Seminario Matematico e Fisico Universita di Modena, vol. 45, no. 2, pp. 527–553, 1997.
- N. Etemadi, “An elementary proof of the strong law of large numbers,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete "Springer", vol. 55, no. 1, pp. 119–122, 1981.
- F. Ezzaki, “SLLN for adapted sequences,” Vietnam Journal of Mathematics, vol. 26, no. 4, pp. 315–322, 1998.
- C. Hess, “Multivalued strong laws of large numbers in the slice topology. Application to integrands,” Set-Valued Analysis, vol. 2, no. 1-2, pp. 183–205, 1994.
- F. Hiai, “Strong laws of large numbers for multivalued random variables,” in Multifunctions and Integrands, pp. 160–172, Springer, 1984.
- F. Akhiat, C. Castaing, and F. Ezzaki, “Some various convergence results for multivalued martingales,” in Advances in Mathematical Economics, vol. 13, pp. 1–33, Springer, 2010.
- T. Chowdhury, Convergence Theorems for Pettis Integrable Functions and Multifunctions, [Ph. D. thesis], University of Burdwan, 2012.
- K. El Amri and C. Hess, “On the Pettis integral of closed valued multifunctions,” Set-Valued Analysis, vol. 8, no. 4, pp. 329–360, 2000.
- R. F. Geitz, “Pettis integration,” Proceedings of the American Mathematical Society, vol. 82, no. 1, pp. 81–86, 1981.
- C. Godet-Thobie and B. Satco, “Decomposability and uniform integrability in Pettis integration,” Quaestiones Mathematicae, vol. 29, no. 1, pp. 39–58, 2006.
- M. Musial, “Martingales of Pettis integrable functions,” in Measure Theory Oberwolfach, pp. 324–339, Springer, 1980.
- C. Castaing and M. Valadier, “Convex analysis and measurable multifunctions,” in Bulletin of the American Mathematical Society, vol. 84, pp. 950–956, American Mathematical Society, 1978.
- F. Akhiat, M. El Harami, and F. Ezzaki, “Pettis conditional expectation of closed convex random sets in a Banach space without RNP,” Journal of the Korean Mathematical Society, vol. 55, no. 4, pp. 833–848, 2018.
- A. Choukairi-Dini, “Sur la convergence d’esperance conditionnelle de multi applications,” in TSAC Montpellier, Expose, vol. 10, 1987.
- P. A. Meyer, “Probabilites et potentiel,” Blaisdell, vol. 14, 1966.
- M. El Harami and F. Ezzaki, “General Pettis conditional expectation and convergence theorems,” International Journal of Mathematics and Statistics, vol. 11, no. 1, pp. 91–111, 2012.
- J. Neveu, Bases Mathematiques Du Calcul Des Probabilites, Masson, 1964.
Copyright © 2019 Hamid Oulghazi and Fatima Ezzaki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.