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 $[0, ∞)$, we study the limits of the ratios of tails $\overline{F^{*\tau}}(x)/\overline{F}(x)$ as $x→∞$ (here, $τ$ is a counting random variable which does not depend on $\{ξ_n\}_{n≥1})$. We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes.