# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Strong limit theorems for blockwise $m$-dependent and blockwise quasi-orthogonal sequences of random variablesHTML articles powered by AMS MathViewer

by F. Móricz
Proc. Amer. Math. Soc. 101 (1987), 709-715 Request permission

## 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.
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