A Weak Law of Large Numbers for Dependent Random Variables

Abstract

Every sequence $f_1,f_2,\cdots$ of random variables with ${\lim }_{M\to \infty }(M{\sup }_{k\in \mathbb{N}}\mathbb{P}(|f_k|>M))=0$ contains a subsequence $f_{k_1},f_{k_2},\cdots$ that satisfies, along with all its subsequences, the weak law of large numbers: ${\lim }_{N\to \infty }((1/N){\sum }_{n=1}^N{f}_{k_n}-D_N)=0,$ in probability. Here $D_N$ is a "corrector" random variable with values in $[-N,N]$, for each $N\in \mathbb{N}$; these correctors are all equal to zero if, in addition, ${\mathrm{liminf}}_{k\to \infty }\mathbb{E}\left(f_k^2{1}_{\{|f_k|\le M\}}\right)=0$ holds for every $M\in (0,\infty ).$