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

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.