## The Annals of Probability

- Ann. Probab.
- Volume 16, Number 4 (1988), 1644-1664.

### 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.

#### Article information

**Source**

Ann. Probab., Volume 16, Number 4 (1988), 1644-1664.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991589

**Digital Object Identifier**

doi:10.1214/aop/1176991589

**Mathematical Reviews number (MathSciNet)**

MR958208

**Zentralblatt MATH identifier**

0655.60015

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60E15: Inequalities; stochastic orderings

Secondary: 60G50: Sums of independent random variables; random walks

**Keywords**

Random sums coarseness Berry-Esseen theorems top $K$ set top $K$ probability sampling stochastic monotonicity log concavity

#### Citation

Fill, James Allen. Bounds on the Coarseness of Random Sums. Ann. Probab. 16 (1988), no. 4, 1644--1664. doi:10.1214/aop/1176991589. https://projecteuclid.org/euclid.aop/1176991589