Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The strong law of large numbers: a weak-$l_ 2$ view
HTML articles powered by AMS MathViewer

by Bernard Heinkel PDF
Proc. Amer. Math. Soc. 123 (1995), 273-280 Request permission

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.
References
Similar Articles
Additional 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