On the maximum of a perturbed random walk

Let (ξ1,η1),(ξ2,η2),… be a sequence of i.i.d. two-dimensional random vectors. We prove a functional limit theorem for the maximum of a perturbed random walk max0≤k≤n(ξ1+⋯+ξk+ηk+1) in a situation where its asymptotics is affected by both max0≤k≤n(ξ1+⋯+ξk) and max1≤k≤nηk to a comparable extent. This solves an open problem that we learned from the paper “Renorming divergent perpetuities” by P. Hitczenko and J. Wesołowski.