Modern Stochastics: Theory and Applications

In this paper, the asmptotics is considered for the distribution tail of a randomly stopped sum ${S_{\nu }}={X_{1}}+\cdots +{X_{\nu }}$ of independent identically distributed consistently varying random variables with zero mean, where ν is a counting random variable independent of $\{{X_{1}},{X_{2}},\dots \}$. The conditions are provided for the relation $\mathbb{P}({S_{\nu }}\gt x)\sim \mathbb{E}\nu \hspace{0.1667em}\mathbb{P}({X_{1}}\gt x)$ to hold, as $x\to \infty $, involving the finiteness of $\mathbb{E}|{X_{1}}|$. The result improves that of Olvera-Cravioto [14], where the finiteness of a moment $\mathbb{E}|{X_{1}}{|^{r}}$ for some $r\gt 1$ was assumed.

Let $\{\xi _{1},\xi _{2},\dots \}$ be a sequence of independent random variables, and η be a counting random variable independent of this sequence. We consider conditions for $\{\xi _{1},\xi _{2},\dots \}$ and η under which the distribution function of the random sum $S_{\eta }=\xi _{1}+\xi _{2}+\cdots +\xi _{\eta }$ belongs to the class of consistently varying distributions. In our consideration, the random variables $\{\xi _{1},\xi _{2},\dots \}$ are not necessarily identically distributed.