Randomly stopped maximum and maximum of sums with consistently varying distributions

Abstract

Let $\{{\xi }_1,{\xi }_2,\dots \}$ be a sequence of independent random variables, and $\eta$ 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 $n⩾1$. We consider conditions for random variables $\{{\xi }_1,{\xi }_2,\dots \}$ and $\eta$ 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.

Published at http://dx.doi.org/10.15559/17-VMSTA74 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)

References in corpus (2)

• Randomly stopped sums with consistently varying distributions
• Random convolution of inhomogeneous distributions with $\mathcal{O}$-exponential tail