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

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

2007-01-31

arxiv.org/abs/math/0701920v3

Bernoulli 2008, Vol. 14, No. 2, 391-404

Mathematics

Probability

Published in at http://dx.doi.org/10.3150/07-BEJ111 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/07-BEJ111

For a distribution $F^{*\tau}$ 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 $\bar{F^{*\tau}}(x)/\bar{F}(x)$ as $x\to\infty$ (here, $\tau$ is a counting random variable which does not depend on $\{\xi_n\}_{n\ge1}$). We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes.

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