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.
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