Lower limits and equivalences for convolution tails

Abstract

Suppose $F$ is a distribution on the half-line $[0,∞)$. We study the limits of the ratios of tails $\overline{F*F}(x)/\overline F(x)$ as $x→∞$. We also discuss the classes of distributions ${\mathscr{S}}$, ${\mathscr{S}}(\gamma)$ and ${\mathscr{S}}^{*}$.