A note on randomly stopped sums with zero mean increments
Volume 11, Issue 1 (2024), pp. 31–42
Pub. online: 5 December 2023
Type: Research Article
Open Access
Open Access
Received
22 September 2023
22 September 2023
Revised
29 October 2023
29 October 2023
Accepted
3 November 2023
3 November 2023
Published
5 December 2023
5 December 2023
Abstract
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.
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