A remark on the paper "Renorming divergent perpetuities"

Abstract

Let $({\xi }_k)$ and $({\eta }_k)$ be infinite independent samples from different distributions. We prove a functional limit theorem for the maximum of a perturbed random walk $\underset{0\le k\le n}{\max }({\xi }_1+\dots +{\xi }_k+{\eta }_{k+1})$ in a situation where its asymptotics is affected by both $\underset{0\le k\le n}{\max }({\xi }_1+\dots +{\xi }_k)$ and $\underset{1\le k\le n}{\max }{\eta }_k$ to a comparable extent. This solves an open problem that we learned from the paper "Renorming divergent perpetuities" by P. Hitczenko and J. Wesołowski.

arXiv comment6 pages