The Annals of Probability

Comparing the Tail of an Infinitely Divisible Distribution with Integrals of its Levy Measure

Paul Embrechts and Charles M. Goldie

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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.

Article information

Source
Ann. Probab., Volume 9, Number 3 (1981), 468-481.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176994419

Digital Object Identifier
doi:10.1214/aop/1176994419

Mathematical Reviews number (MathSciNet)
MR614631

Zentralblatt MATH identifier
0459.60017

JSTOR
links.jstor.org

Subjects
Primary: 60E07: Infinitely divisible distributions; stable distributions

Keywords
Generalised gamma convolutions infinite divisibility Levy measures rapid variation regular variation subexponentiality tails of probability distributions

Citation

Embrechts, Paul; Goldie, Charles M. Comparing the Tail of an Infinitely Divisible Distribution with Integrals of its Levy Measure. Ann. Probab. 9 (1981), no. 3, 468--481. doi:10.1214/aop/1176994419. https://projecteuclid.org/euclid.aop/1176994419


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