Note on Completely Monotone Densities

F.W. Sleutel

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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 languageEnglish
Pages (from-to)1130-1131
JournalAnnals of Mathematical Statistics
Volume40
Issue number3
DOIs
Publication statusPublished - 1969

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Completely Monotone
Infinitely Divisible
Completely Monotone Function
Monotone Sequences
Geometric distribution
Exponential distribution
Density Function
Analogue
Imply
Integer

Keywords

  • IR-70395

Cite this

Sleutel, F.W. / Note on Completely Monotone Densities. In: Annals of Mathematical Statistics. 1969 ; Vol. 40, No. 3. pp. 1130-1131.
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Note on Completely Monotone Densities. / Sleutel, F.W.

In: Annals of Mathematical Statistics, Vol. 40, No. 3, 1969, p. 1130-1131.

Research output: Contribution to journalArticleAcademic

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AB - 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].

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