Bounds on the Coarseness of Random Sums

Abstract

Let $X^0_1, \ldots, X^0_N$ be integer-valued random variables and let $a_1, \ldots, a_N$ be (fixed) nonzero vectors. We introduce the notion of coarseness of a discrete distribution and obtain upper bounds on the coarseness of the distribution of $S = \Sigma X^0_ia_i$ by comparison with the case $a_i \equiv a$. The bounds derived are seen to be tight and to apply for example when $S$ is formed (a) from independent summands or (b) by using any of a large family of sampling schemes. We show how such bounds can easily and efficiently substitute for use of Berry-Esseen theorems and other analytical methods in applications.