Forward-convex convergence in probability of sequences of nonnegative random variables

Kardaras, Constantinos; Žitković, Gordan

doi:10.1090/S0002-9939-2012-11373-5

Forward-convex convergence in probability of sequences of nonnegative random variables
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by Constantinos Kardaras and Gordan Žitković PDF

Proc. Amer. Math. Soc. 141 (2013), 919-929 Request permission

Abstract:

For a sequence $(f_n)_{n \in \mathbb {N}}$ of nonnegative random variables, we provide simple necessary and sufficient conditions for convergence in probability of each sequence $(h_n)_{n \in \mathbb {N}}$ with $h_n\in \mathrm {conv}(\{f_n,f_{n+1},\dots \})$ for all $n \in \mathbb {N}$ to the same limit. These conditions correspond to an essentially measure-free version of the notion of uniform integrability.

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