Modern Stochastics: Theory and Applications

Author: Martynas Manstavičius

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Randomly stopped maximum and maximum of sums with consistently varying distributions

Ieva Marija Andrulytė Martynas Manstavičius Jonas Šiaulys

https://doi.org/10.15559/17-VMSTA74

Pub. online: 6 Mar 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 1 (2017), pp. 65–78

Abstract

Let

$\{\xi _{1},\xi _{2},\dots \}$ be a sequence of independent random variables, and η be a counting random variable independent of this sequence. In addition, let

$S_{0}:=0$ and

$S_{n}:=\xi _{1}+\xi _{2}+\cdots +\xi _{n}$ for

$n\geqslant 1$ . We consider conditions for random variables

$\{\xi _{1},\xi _{2},\dots \}$ and η under which the distribution functions of the random maximum

$\xi _{(\eta )}:=\max \{0,\xi _{1},\xi _{2},\dots ,\xi _{\eta }\}$ and of the random maximum of sums

$S_{(\eta )}:=\max \{S_{0},S_{1},S_{2},\dots ,S_{\eta }\}$ belong to the class of consistently varying distributions. In our consideration the random variables

$\{\xi _{1},\xi _{2},\dots \}$ are not necessarily identically distributed.

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