Randomly stopped maximum and maximum of sums with consistently varying distributions

• Main Authors: Ieva Marija Andrulytė, Martynas Manstavičius, Jonas Šiaulys
• Format: Article
• Language: English
• Published: VTeX 2017-03-01
• Series: Modern Stochastics: Theory and Applications
• Subjects: Heavy tail; consistently varying tail; randomly stopped maximum; randomly stopped maximum of sums; closure property
• Online Access: https://vmsta.vtex.vmt/doi/10.15559/17-VMSTA74

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.