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} \big( M \sup_{k \in \mathbb{N}} \mathbb{P} ( |f_k| > M ) \big)=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} \big( (1/N) \sum_{n=1}^N f_{k_n} - D_N \big) =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, $\, \liminf_{k \to \infty} \mathbb{E} \big( f_k^2 \, \mathbf{ 1}_{ \{ |f_k| \le M \} } \big) =0\,$ holds for every $M \in (0, \infty)\,.$