### 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].

Original language | English |
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Pages (from-to) | 1130-1131 |

Journal | Annals of Mathematical Statistics |

Volume | 40 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1969 |

### Keywords

- IR-70395

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## Cite this

Sleutel, F. W. (1969). Note on Completely Monotone Densities.

*Annals of Mathematical Statistics*,*40*(3), 1130-1131. https://doi.org/10.1214/aoms/1177697626