## The Annals of Probability

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

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

Paul Embrechts and Charles M. Goldie

#### 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