Lower limits and equivalences for convolution tails

Suppose $F$ is a distribution on the half-line $[0,\mathrm{\infty })$. We study the limits of the ratios of tails $\overline{F\ast F}(x)\mathrm{/}\bar{F}(x)$ as $x\to \mathrm{\infty }$. We also discuss the classes of distributions $\mathcal{S}$, $\mathcal{S}(\gamma )$ and ${\mathcal{S}}^{\ast }$.