A remark on the paper "Renorming divergent perpetuities"

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 $\underset{0\leq k\leq n}{\max}\,(\xi_1+\ldots+\xi_k+\eta_{k+1})$ in a situation where its asymptotics is affected by both $\underset{0\leq k\leq n}{\max}\,(\xi_1+\ldots+\xi_k)$ and $\underset{1\leq k\leq n}{\max}\,\eta_k$ to a comparable extent. This solves an open problem that we learned from the paper "Renorming divergent perpetuities" by P. Hitczenko and J. Weso{\l}owski.