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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 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The strong law of large numbers: a weak-$l_ 2$ view
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by Bernard Heinkel
Proc. Amer. Math. Soc. 123 (1995), 273-280
DOI: https://doi.org/10.1090/S0002-9939-1995-1213860-X

Abstract:

Let $({X_k})$ be a sequence of independent, centered, and square integrable real-valued random variables. To that sequence one associates \[ \forall n \in \mathbb {N},\quad {\xi _n} = {\left \| {({2^{ - n}}{X_k}),\;{2^n} + 1 \leq k \leq {2^{n + 1}}} \right \|_{2,\infty }}.\] When there exists $K \geq 1$ such that \[ \sum \limits _{n \geq 1} {{P^K}({\xi _n} > {c_n}) < + \infty ,} \] 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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Bibliographic Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 273-280
  • MSC: Primary 60F15; Secondary 46B45, 46N30, 60E15, 60G50
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1213860-X
  • MathSciNet review: 1213860