## The Annals of Mathematical Statistics

### Note on Completely Monotone Densities

F. W. Steutel

#### 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

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