On lower limits and equivalences for distribution tails of randomly stopped sums

Abstract

For a distribution $F^{\ast \tau }$ of a random sum $S_{\tau }={\xi }_1+...+{\xi }_{\tau }$ of i.i.d. random variables with a common distribution $F$ on the half-line $[0,\infty )$, we study the limits of the ratios of tails $\overline{F^{\ast \tau }}(x)/\bar{F}(x)$ as $x\to \infty$ (here, $\tau$ is a counting random variable which does not depend on $\{{\xi }_n{\}}_{n\ge 1}$). We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes.

Published in at http://dx.doi.org/10.3150/07-BEJ111 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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