A renewal theorem and supremum of a perturbed random walk

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.

12 pages. arXiv admin note: substantial text overlap with arXiv:1701.02625

References in corpus (3)

• Tail asymptotics for the maximum of perturbed random walk
• Renorming divergent perpetuities
• The left tail of renewal measure

Cited by in corpus (2)

• Stochastic recursions: between Kesten's and Grincevičius-Grey's assumptions
• A result on power moments of Lévy-type perpetuities and its application to the $L_p$-convergence of Biggins' martingales in branching Lévy processes