Modern Stochastics: Theory and Applications

Keywords: 60G40

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A note on randomly stopped sums with zero mean increments

Remigijus Leipus Jonas Šiaulys

https://doi.org/10.15559/23-VMSTA236

Pub. online: 5 Dec 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 1 (2024), pp. 31–42

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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