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].
Citation
F. W. Steutel. "Note on Completely Monotone Densities." Ann. Math. Statist. 40 (3) 1130 - 1131, June, 1969. https://doi.org/10.1214/aoms/1177697626
Information
