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Abstract Let (ξ 1, η 1),(ξ 2, η 2),… be a sequence of iid two-dimensional random vectors. We prove a functional limit theorem for the maximum of a perturbed random walk max 0≤ k≤ n (ξ 1+⋯+ ξ k+ η k+ 1) in a situation where its asymptotics is affected by both max 0≤ k≤ n (ξ 1+⋯+ ξ k) and max 1≤ k≤ n η 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.