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We study tails of the supremum of a perturbed random walk under regime whichwas not yet considered in the literature. Our approach is based on a newrenewal theorem, which is of independent interest. We obtain first and secondorder asymptotics of the solution to renewal equation under weak assumptionsand we apply these results to obtain first and second order asymptotics of thetail of the supremum of a perturbed random walk.

A renewal theorem and supremum of a perturbed random walk