### Abstract

In this note the asymptotic behaviour of absolute moments of infinitely divisible (inf div) distributions is considered. It follows from the fact that tails of inf div distributions are bounded from below that their moments are bounded from below. It turns out that moments of inf div distributions also behave more regularly than moments in general, and a strong similarity is shown to exist between the moments of inf div distributions on the half-line and the probabilities of inf div distributions on the nonnegative integers. Regularity properties of the moment generating function \phi are used to show that limits of the form log \alpha_n / (nk(n)) exist, where \alpha_n = \beta_n / n! with \beta_n the n-th absolute moment. Furthermore, it is shown that the boundedness of the Poisson spectrum has a simple characterization in terms of moment behaviour. Finally, this kind of behaviour is related to the order and type of the functions \phi and log \phi.
Key words and phrases: Infinitely divisible, (absolute) moments, asymptotic behaviour, tails of distribution functions, moment generating function, characteristic function, entire function.

Original language | English |
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Place of Publication | Eindhoven |

Publisher | Technische Hogeschool Eindhoven |

Number of pages | 13 |

Publication status | Published - 1977 |

### Publication series

Name | Memorandum COSOR |
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Volume | 7706 |

ISSN (Print) | 0926-4493 |

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## Cite this

Steutel, F. W., & Wolfe, S. J. (1977).

*On the asymptotic behaviour of moments of infinitely dicisible distributions*. (Memorandum COSOR; Vol. 7706). Technische Hogeschool Eindhoven.