A Strong Limit Theorem for Double Arrays of Dependent Random Variables

Abstract

For a double array of random variables $\{X_{m,n},\,m\geq 1,n\geq 1\}$ such that for any $k\geq 1,$ $\ell\geq 1$, the collection $\{X_{i,j},2^{k-1}\leq i<2^{k},2^{\ell-1}\leq i<2^{\ell}\}$ is $M$-pairwise negatively dependent, conditions are provided under which $\max\limits_{k\leq m,\,\ell\leq n}\left|\sum_{i=1}^{k}\sum_{j=1}^{\ell}X_{i,j}\right|/(a_{m}b_{n})\to 0$ almost surely as $\max(m,\,n)\to\infty$, where $\{a_{n},\,n\geq 1\}$ and $\{b_{n},\,n\geq 1\}$ are two sequences of positive constants. The sharpness of the result is illustrated by an example.