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August, 1977 A Strong Law for Weighted Averages of Independent, Identically Distributed Random Variables with Arbitrarily Heavy Tails
F. T. Wright, Ronald D. Platt, Tim Robertson
Ann. Probab. 5(4): 586-590 (August, 1977). DOI: 10.1214/aop/1176995767

Abstract

Let $X_1, X_2, \cdots$ be independent, identically distributed, nondegenerate random variables, let $w_k$ be a sequence of positive numbers and for $n = 1,2, \cdots$ let $S_n = \sum^n_{k=1} w_kX_k$ and $W_n = \sum^n_{k=1} w_k$. The weak (strong) law is said to hold for $\{X_k, w_k\}$ if and only if $S_n/W_n$ converges in probability (almost surely) to a constant. Jamison, Orey and Pruitt (1965) (Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 40-44) studied conditions related to these laws of large numbers. In considering the strong law, only distributions with finite first moments are discussed. However, Theorem 2 of this paper shows that a sequence of random variables and a sequence of weights can be chosen so that the strong law holds and so that the random variables have arbitrarily heavy tails. This result also answers some interesting questions concerning the weak law.

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F. T. Wright. Ronald D. Platt. Tim Robertson. "A Strong Law for Weighted Averages of Independent, Identically Distributed Random Variables with Arbitrarily Heavy Tails." Ann. Probab. 5 (4) 586 - 590, August, 1977. https://doi.org/10.1214/aop/1176995767

Information

Published: August, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0367.60029
MathSciNet: MR494436
Digital Object Identifier: 10.1214/aop/1176995767

Subjects:
Primary: 60F15
Secondary: 60F05

Rights: Copyright © 1977 Institute of Mathematical Statistics

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Vol.5 • No. 4 • August, 1977
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