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Let $(\xi_k)$ and $(\eta_k)$ be infinite independent samples from differentdistributions. We prove a functional limit theorem for the maximum of aperturbed random walk $\underset{0\leq k\leqn}{\max}\,(\xi_1+\ldots+\xi_k+\eta_{k+1})$ in a situation where its asymptoticsis 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 anopen problem that we learned from the paper "Renorming divergent perpetuities"by P. Hitczenko and J. Weso{\l}owski.

A remark on the paper "Renorming divergent perpetuities"