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

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.