# A renewal theorem and supremum of a perturbed random walk

## TL;DR

This paper introduces a new renewal theorem to analyze the tail behavior of the supremum of a perturbed random walk, providing novel asymptotic results under weak assumptions.

## Contribution

It develops a new renewal theorem and applies it to derive first and second order asymptotics for the tail of the supremum of a perturbed random walk, a regime not previously studied.

## Key findings

- Established first and second order asymptotics for the tail of the supremum
- Developed a new renewal theorem of independent interest
- Extended analysis to a previously unconsidered regime

## Abstract

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