## The Annals of Probability

- Ann. Probab.
- Volume 13, Number 1 (1985), 299-306.

### On the Unimodality of High Convolutions of Discrete Distributions

A. M. Odlyzko and L. B. Richmond

#### Abstract

It is shown that if $\{p_j\}$ is a discrete density function on the integers with support contained in $\{0, 1, \cdots, d\}$, and $p_0 > 0, p_1 > 0, p_{d - 1} > 0, p_d > 0$, then there is an $n_0$ such that the $n$-fold convolution $\{p_j\}^{\ast_n}$ is unimodal for all $n \geq n_0$. Examples show that this result is nearly best possible, but weaker results are proved under less restrictive assumptions.

#### Article information

**Source**

Ann. Probab., Volume 13, Number 1 (1985), 299-306.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993082

**Digital Object Identifier**

doi:10.1214/aop/1176993082

**Mathematical Reviews number (MathSciNet)**

MR770644

**Zentralblatt MATH identifier**

0561.60021

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60E05: Distributions: general theory

**Keywords**

Unimodality discrete distributions

#### Citation

Odlyzko, A. M.; Richmond, L. B. On the Unimodality of High Convolutions of Discrete Distributions. Ann. Probab. 13 (1985), no. 1, 299--306. doi:10.1214/aop/1176993082. https://projecteuclid.org/euclid.aop/1176993082