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September, 1955 Bounds for the Distribution Function of a Sum of Independent, Identically Distributed Random Variables
Wassily Hoeffding, S. S. Shrikhande
Ann. Math. Statist. 26(3): 439-449 (September, 1955). DOI: 10.1214/aoms/1177728489

Abstract

The problem is considered of obtaining bounds for the (cumulative) distribution function of the sum of $n$ independent, identically distributed random variables with $k$ prescribed moments and given ranger. For $n = 2$ it is shown that the best bounds are attained or arbitrarily closely approach with discrete random varibles which take on at most $2k + 2$ values. For nonnegative random variables with given mean, explicit bounds are obtained when $n = 2$; for arbitrary values of $n$, bounds are given which are asymptotically best in the "tail" of the distribution. Some of the results contribute to the more general problem of obtaining bounds for the expected values of a given function of independent, identically distributed random variables when the expected values of certain functions of the individual variables are given. Although the results are modest in scope, the authors hope that the paper will draw attention to a problem of both mathematical and statistical interest.

Citation

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Wassily Hoeffding. S. S. Shrikhande. "Bounds for the Distribution Function of a Sum of Independent, Identically Distributed Random Variables." Ann. Math. Statist. 26 (3) 439 - 449, September, 1955. https://doi.org/10.1214/aoms/1177728489

Information

Published: September, 1955
First available in Project Euclid: 28 April 2007

zbMATH: 0066.37504
MathSciNet: MR72377
Digital Object Identifier: 10.1214/aoms/1177728489

Rights: Copyright © 1955 Institute of Mathematical Statistics

Vol.26 • No. 3 • September, 1955
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