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February, 1980 Necessary and Sufficient Conditions for Complete Convergence in the Law of Large Numbers
Soren Asmussen, Thomas G. Kurtz
Ann. Probab. 8(1): 176-182 (February, 1980). DOI: 10.1214/aop/1176994835

Abstract

Relationships between the growth of a sequence $N_k$ and conditions on the tail of the distribution of a sequence $X_l$ of i.i.d. mean zero random variables are given that are necessary and sufficient for $$\sum^\infty_{k=1} P\big\{\big|\frac{1}{N_k} \sum^{N_k}_{l=1} X_l\big| > \varepsilon\big\} < \infty.$$ The results are significant for distributions satisfying $E(|X_l|) < \infty$ but $E(|X_l|^\beta) = \infty$ for some $\beta > 1$. Necessary and sufficient conditions for the finiteness of sums of the form $$\sum^\infty_{n=1} \gamma (n)P\big\{\big|\frac{1}{n} \sum^n_{l=1} X_l\big| > \varepsilon\big\}$$ are obtained as a corollary.

Citation

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Soren Asmussen. Thomas G. Kurtz. "Necessary and Sufficient Conditions for Complete Convergence in the Law of Large Numbers." Ann. Probab. 8 (1) 176 - 182, February, 1980. https://doi.org/10.1214/aop/1176994835

Information

Published: February, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0426.60026
MathSciNet: MR556425
Digital Object Identifier: 10.1214/aop/1176994835

Subjects:
Primary: 60F10
Secondary: 60F05, 60F15

Rights: Copyright © 1980 Institute of Mathematical Statistics

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Vol.8 • No. 1 • February, 1980
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