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June, 1981 Comparing the Tail of an Infinitely Divisible Distribution with Integrals of its Levy Measure
Paul Embrechts, Charles M. Goldie
Ann. Probab. 9(3): 468-481 (June, 1981). DOI: 10.1214/aop/1176994419

Abstract

Let $F$ be an infinitely divisible distribution on $\lbrack 0, \infty)$, with Levy measure $\nu$. For all real $r$, define measures $\nu_r$ by $\nu_r(dx) = x^r\nu(dx) (x > 1), = 0 (x \leq 1)$. For $0 < \alpha < \infty$, and $- \infty < r' < \alpha < r < \infty$, it is proved that $\nu_{r'}(x, \infty)$ is regularly varying (at $\infty$) with exponent $r' - \alpha$ if and only if $1 - F$ is regularly varying with exponent $- \alpha$ if and only if $\nu_r(0, x\rbrack$ is regularly varying with exponent $r - \alpha.$ If any of this is the case there follow asymptotic relations between $1 - F$ and either of $\nu_{r'}(x, \infty)$ or $\nu_r(0, x\rbrack$. The paper characterises those distributions for which these asymptotic relations hold, some of the characterisations being complete and others assuming that not all moments of $F$ are finite. The characterising classes involve regular variation, second order (de Haan) regular variation, rapid variation, and subexponentiality. An intermediate result is that when $F$ has finite $n$th and infinite $(n + 1)$th moment, $\int^t_0 x^{n + 1}\{1 - F(x)\} dx \sim \int^t_0 x^{n + 1}\nu(x, \infty) dx$ as $t \rightarrow \infty$. The results are applied to generalised gamma convolutions.

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Paul Embrechts. Charles M. Goldie. "Comparing the Tail of an Infinitely Divisible Distribution with Integrals of its Levy Measure." Ann. Probab. 9 (3) 468 - 481, June, 1981. https://doi.org/10.1214/aop/1176994419

Information

Published: June, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0459.60017
MathSciNet: MR614631
Digital Object Identifier: 10.1214/aop/1176994419

Subjects:
Primary: 60E07

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 3 • June, 1981
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