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A Lévy Type Martingale Convergence Theorem for Random Sets with Unbounded Values

Author

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  • Jérôme Couvreux

    (Université Paris Dauphine)

  • Christian Hess

    (Université Paris Dauphine)

Abstract

Given a nondecreasing sequence (ℬ n ) of sub-σ-fields and a real or vector valued random variable f, the Lévy Martingale convergence Theorem (LMCT) asserts that E(f/ℬ n ) converges to E(f/ℬ∞) almost surely and in L 1, where ℬ∞ stands for the σ-field generated by the ℬ n . In the present paper, we study the validity of the multivalued analog this theorem for a random set F whose values are members of ℱ(X), the space of nonempty closed sets of a Banach space X, when ℱ(X) is endowed either with the Painlevé–Kuratowski convergence or its infinite dimensional extensions. We deduce epi-convergence results for integrands via the epigraphical multifunctions. As it is known, these results are useful for approximating optimization problems. The method relies on countability supportness hypotheses which are shown to hold when the values of the random set E(F/ℬ n ) do not contain any line. On the other hand, since the values of F are not assumed to be bounded, conditions involving barrier and asymptotic cones are shown to be necessary. Moreover, we discuss the relations with other multivalued martingale convergence theorems and provide examples showing the role of the hypotheses. Even in the finite dimensional setting, our results are new or subsume already existing ones.

Suggested Citation

  • Jérôme Couvreux & Christian Hess, 1999. "A Lévy Type Martingale Convergence Theorem for Random Sets with Unbounded Values," Journal of Theoretical Probability, Springer, vol. 12(4), pages 933-969, October.
    • Handle: RePEc:spr:jotpro:v:12:y:1999:i:4:d:10.1023_a:1021688919194
    DOI: 10.1023/A:1021688919194
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    References listed on IDEAS

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    1. Hess, Christian, 1991. "On multivalued martingales whose values may be unbounded: martingale selectors and mosco convergence," Journal of Multivariate Analysis, Elsevier, vol. 39(1), pages 175-201, October.
    2. Christian Hess, 1991. "Convergence of Conditional Expectations for Unbounded Random Sets, Integrands, and Integral Functionals," Mathematics of Operations Research, INFORMS, vol. 16(3), pages 627-649, August.
    3. Hiai, Fumio & Umegaki, Hisaharu, 1977. "Integrals, conditional expectations, and martingales of multivalued functions," Journal of Multivariate Analysis, Elsevier, vol. 7(1), pages 149-182, March.
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    Cited by:

    1. Pedro Terán, 2016. "A Multivalued Strong Law of Large Numbers," Journal of Theoretical Probability, Springer, vol. 29(2), pages 349-358, June.

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