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

 

SLLN and convergence rates for nearly orthogonal sequences of random variables


Author: Ferenc Móricz
Journal: Proc. Amer. Math. Soc. 95 (1985), 287-294
MSC: Primary 60F15; Secondary 60G48
MathSciNet review: 801340
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Abstract: Let $ \{ {X_k}:k \geqslant 1\} $ be a sequence of random variables with finite second moments $ EX_k^2 = \sigma _k^2 < \infty $ for which $ \vert E{X_k}{X_l}\vert \leqslant {\sigma _k}{\sigma _l}\rho (\vert k - l\vert)$, where $ \{ \rho (j):j \geqslant 0\} $ is a sequence of nonnegative numbers with $ \sum\nolimits_{j = 0}^\infty {\rho (j)} < \infty $. In particular, in the case of orthogonality, $ \rho (j) = 0$ for $ j \geqslant 1$. We prove strong laws for the first arithmetic means $ {\zeta _n} = {n^{ - 1}}\sum\nolimits_{k = 1}^n {{X_k}} $ and the Cesàro means

$\displaystyle {\tau _n} = {n^{ - 1}}\sum\limits_{k = 1}^n {(1 - (k - 1){n^{ - 1}}){X_k}} .$

The convergence rates are studied in the form $ P\{ {\sup _{n > {2^p}}}\vert{\zeta _n}\vert > \varepsilon \} $ and $ P\{ {\sup _{n > {2^p}}}\vert{\tau _n}\vert > \varepsilon \} $, where $ \varepsilon > 0$ is fixed and $ p$ tends to $ \infty $. At the end, the case where $ \Sigma _{j = 0}^\infty \rho (j) = \infty $ is also treated.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0801340-6
PII: S 0002-9939(1985)0801340-6
Keywords: Orthogonal and quasi-orthogonal random variables, first arithmetic means, Cesàro means, strong laws of large numbers, rates of convergence
Article copyright: © Copyright 1985 American Mathematical Society