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

Abstract

For a distribution $F^{*τ}$ of a random sum $S_τ=ξ_1+...+ξ_τ$ 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 $\bar{F^{*τ}}(x)/\bar{F}(x)$ as $x\to\infty$ (here, $τ$ is a counting random variable which does not depend on $\{ξ_n\}_{n\ge1}$). We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes.