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

Jérôme Couvreux and Christian Hess ()
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Jérôme Couvreux: Université Paris Dauphine
Christian Hess: Université Paris Dauphine

Journal of Theoretical Probability, 1999, vol. 12, issue 4, 933-969

Abstract: 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.

Keywords: Measurable multifunctions; random sets; multivalued conditional expectations; multivalued martingales; set convergences; epi-convergence; integrands; asymptotic cone (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1023/A:1021688919194

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