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December, 1977 Iterated Logarithm Laws for Asymmetric Random Variables Barely with or Without Finite Mean
Michael Klass, Henry Teicher
Ann. Probab. 5(6): 861-874 (December, 1977). DOI: 10.1214/aop/1176995656

Abstract

One-sided iterated logarithm laws of the form $\lim \sup (1/b_n) \sum^n_1 X_i = 1$, a.s. and $\lim \sup (1/b_n) \sum^n_1 X_i = -1$, a.s. are obtained for asymmetric independent and identically distributed random variables, the first when these have a vanishing but barely finite mean, the second when $E|X|$ is barely infinite. In both cases, $\lim \inf (1/b_n) \sum^n_1 X_i = -\infty$, a.s. The constants $b_n/n$ are slowly varying, decreasing to zero in the first case and increasing to infinity in the second. Although defined via the distribution of $|X|, b_n$ represents the order of magnitude of $E|\sum^n_1 X_i|$ when this is finite. Corresponding weak laws of large numbers are established and related to Feller's notion of "unfavorable fair games" and in the process a theorem playing the same role for the weak law as Feller's generalization of the strong law is proved.

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Michael Klass. Henry Teicher. "Iterated Logarithm Laws for Asymmetric Random Variables Barely with or Without Finite Mean." Ann. Probab. 5 (6) 861 - 874, December, 1977. https://doi.org/10.1214/aop/1176995656

Information

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

zbMATH: 0372.60042
MathSciNet: MR445588
Digital Object Identifier: 10.1214/aop/1176995656

Subjects:
Primary: 60F15

Rights: Copyright © 1977 Institute of Mathematical Statistics

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Vol.5 • No. 6 • December, 1977
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