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

 

The strong law of large numbers: a weak-$ l\sb 2$ view


Author: Bernard Heinkel
Journal: Proc. Amer. Math. Soc. 123 (1995), 273-280
MSC: Primary 60F15; Secondary 46B45, 46N30, 60E15, 60G50
MathSciNet review: 1213860
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Abstract: Let $ ({X_k})$ be a sequence of independent, centered, and square integrable real-valued random variables. To that sequence one associates

$\displaystyle \forall n \in \mathbb{N},\quad {\xi _n} = {\left\Vert {({2^{ - n}}{X_k}),\;{2^n} + 1 \leq k \leq {2^{n + 1}}} \right\Vert _{2,\infty }}.$

When there exists $ K \geq 1$ such that

$\displaystyle \sum\limits_{n \geq 1} {{P^K}({\xi _n} > {c_n}) < + \infty ,} $

where $ ({c_n})$ is a suitable sequence of positive constants, then the strong law of large numbers holds if and only if $ ({X_k}/k)$ converges almost surely to 0.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1213860-X
PII: S 0002-9939(1995)1213860-X
Keywords: Strong law of large numbers, Rademacher sums, weak-$ {l_p}$ spaces
Article copyright: © Copyright 1995 American Mathematical Society