## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 40, Number 3 (1969), 1130-1131.

### Note on Completely Monotone Densities

#### Abstract

In [2] it is proved that mixtures of exponential distributions are infinitely divisible (id). In [3] it is proved that the same holds for the discrete analogue, i.e. for mixtures of geometric distributions. In this note we show that these results imply that a density function $f(x)$ (or distribution $\{p_n\}$ on the integers) is id if the function $f(x)$ (or the sequence $\{p_n\}$ is completely monotone (cm). For the definition and properties of cm functions and sequences we refer to [1].

#### Article information

**Source**

Ann. Math. Statist., Volume 40, Number 3 (1969), 1130-1131.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177697626

**Digital Object Identifier**

doi:10.1214/aoms/1177697626

**Mathematical Reviews number (MathSciNet)**

MR254949

**Zentralblatt MATH identifier**

0183.47601

**JSTOR**

links.jstor.org

#### Citation

Steutel, F. W. Note on Completely Monotone Densities. Ann. Math. Statist. 40 (1969), no. 3, 1130--1131. doi:10.1214/aoms/1177697626. https://projecteuclid.org/euclid.aoms/1177697626