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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: F. Móricz
Journal: Proc. Amer. Math. Soc. 101 (1987), 709-715
MSC: Primary 60F15; Secondary 60G50
MathSciNet review: 911038
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0911038-3
PII: S 0002-9939(1987)0911038-3
Keywords: $ m$-dependent random variables, orthogonal random variables, first arithmetic means, strong limit theorems, strong laws of large numbers
Article copyright: © Copyright 1987 American Mathematical Society