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Bounds for the Distribution Function of a Sum of Independent, Identically Distributed Random Variables
Wassily Hoeffding and S. S. Shrikhande
The Annals of Mathematical Statistics
Vol. 26, No. 3 (Sep., 1955), pp. 439-449
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2236471
Page Count: 11
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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.
The Annals of Mathematical Statistics © 1955 Institute of Mathematical Statistics