Strong limit theorems for blockwise $m$-dependent and blockwise quasi-orthogonal sequences of random variables

Abstract:

Let $\{ {X_k}:k = 1,2, \ldots \}$ be a sequence of random variables with zero mean and finite variance $\sigma _k^2$. We say that $\{ {X_k}\}$ is blockwise $m$-dependent if for each $p$ large enough the following is true: if we remove $m$ or more consecutive $X$’s from the dyadic block $\{ {X_{{2^{p - 1}} + 1}}, \ldots ,{X_{{2^p}}}\}$, then the two remaining portions are independent. We say that $\{ {X_k}\}$ is blockwise quasiorthogonal if for each $p$, the expectations $E({X_k}{X_l})$ are small in a certain sense again within the dyadic block $\{ {X_{{2^{p - 1}} + 1}}, \ldots ,{X_{{2^p}}}\}$. Blockwise independence and blockwise orthogonality are particular cases of the above notions, respectively. We study the a.s. behavior of the series $\sum \nolimits _{k = 1}^\infty {{X_k}}$ and that of the first arithmetic means $(1/n)\sum \nolimits _{k = 1}^n {{X_k}}$. It turns out that the classical strong limit theorems, with one exception, remain valid in this more general setting, too.