Forward-convex convergence in probability of sequences of nonnegative random variables
Kardaras, C.
& Žitković, G.
(2013).
Forward-convex convergence in probability of sequences of nonnegative random variables.
Proceedings of the American Mathematical Society,
141(3), 919-929.
https://doi.org/10.1090/S0002-9939-2012-11373-5
& Žitković, G.
(2013).
Forward-convex convergence in probability of sequences of nonnegative random variables.
Proceedings of the American Mathematical Society,
141(3), 919-929.
https://doi.org/10.1090/S0002-9939-2012-11373-5
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.