Geodesic
A weak law of large numbers for dependent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 3, pp. 619-629

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $M\in(0,\infty)$.
Keywords: weak law of large numbers, hereditary convergence, weak convergence, truncation, generalized expectation, nonlinear expectation. 
@article{TVP_2023_68_3_a10,
     author = {I. Karatzas and W. Schachermayer},
     title = {A weak law of large numbers for dependent random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {619--629},
     publisher = {mathdoc},
     volume = {68},
     number = {3},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2023_68_3_a10/}
}
TY  - JOUR
AU  - I. Karatzas
AU  - W. Schachermayer
TI  - A weak law of large numbers for dependent random variables
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2023
SP  - 619
EP  - 629
VL  - 68
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TVP_2023_68_3_a10/
LA  - ru
ID  - TVP_2023_68_3_a10
ER  - 
%0 Journal Article
%A I. Karatzas
%A W. Schachermayer
%T A weak law of large numbers for dependent random variables
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2023
%P 619-629
%V 68
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TVP_2023_68_3_a10/
%G ru
%F TVP_2023_68_3_a10
I. Karatzas; W. Schachermayer. A weak law of large numbers for dependent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 3, pp. 619-629. http://geodesic.mathdoc.fr/item/TVP_2023_68_3_a10/