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December, 1972 Inequalities for the Law of Large Numbers
Thomas G. Kurtz
Ann. Math. Statist. 43(6): 1874-1883 (December, 1972). DOI: 10.1214/aoms/1177690858


Let $X_1, X_2, X_3, \cdots$ be independent random variables and $a_1, a_2, a_3, \cdots$ positive real numbers. Define $F(t) = \sup_k P\{|X_k| > t\}$ and $S_m = \sum^m_{k=1} a_k X_k.$ Inequalities of the form $P\{\sup_m|S_m| > \delta\} \leqq C \sum_k \int^1_0 \varphi'(u)F(u/a_k) du$ are given for a large class of functions $\varphi$, as well as inequalities of a somewhat different form that are appropriate for considering exponential convergence rates. Examples of how the inequalities can be used to prove rate theorems are also given.


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Thomas G. Kurtz. "Inequalities for the Law of Large Numbers." Ann. Math. Statist. 43 (6) 1874 - 1883, December, 1972.


Published: December, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0251.60019
MathSciNet: MR378045
Digital Object Identifier: 10.1214/aoms/1177690858

Rights: Copyright © 1972 Institute of Mathematical Statistics

Vol.43 • No. 6 • December, 1972
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