The Annals of Mathematical Statistics

Inequalities for the Law of Large Numbers

Thomas G. Kurtz

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Abstract

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.

Article information

Source
Ann. Math. Statist., Volume 43, Number 6 (1972), 1874-1883.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177690858

Digital Object Identifier
doi:10.1214/aoms/1177690858

Mathematical Reviews number (MathSciNet)
MR378045

Zentralblatt MATH identifier
0251.60019

JSTOR
links.jstor.org

Citation

Kurtz, Thomas G. Inequalities for the Law of Large Numbers. Ann. Math. Statist. 43 (1972), no. 6, 1874--1883. doi:10.1214/aoms/1177690858. https://projecteuclid.org/euclid.aoms/1177690858


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