A weak law of large numbers for dependent random variables

I. Karatzas   ; W. Schachermayer

Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 3, pp. 619-629

Résumé

Each sequence $f_1,f_2,\dots$ of random variables satisfying $\lim_{M\to \infty}(M\sup_{k\in \mathbf N}\mathbf{P}(|f_k|>M))=0$} contains a subsequence $f_{k_1},f_{k_2},\dots$ which, along with all its subsequences, satisfies the weak law of large numbers $\lim_{N\to\infty}\bigl((1/N) \sum^N_{n=1} f_{k_n}- D_N\bigr)=0$ in probability. Here, $D_N$ is a “corrector” random variable with values in $[-N,N]$ for each $N\in\mathbf{N}$; these correctors are all equal to zero if, in addition, $\lim \inf_{n\to\infty}\mathbf{E}(f^2_n \mathbf{1}_{\{|f_n|\le M\}})=0$ for every $M\in(0,\infty)$.

Détail

Keywords: weak law of large numbers, hereditary convergence, weak convergence, truncation, generalized expectation, nonlinear expectation.

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