Local tail asymptotics for the joint distribution of length and of maximum of a random walk excursion

TL;DR

This paper investigates the detailed asymptotic behavior of the joint distribution of the length, maximum, and timing of a random walk excursion with negative drift, providing insights into their local limit properties.

Contribution

It derives the local asymptotics of the joint distribution of excursion length, maximum, and timing for a random walk with negative drift and light-tailed increments, enabling local limit theorems.

Findings

Derived local asymptotics for joint distribution

Established local central limit theorems conditioned on large maxima

Provided detailed probabilistic descriptions of excursion characteristics

Abstract

This note is devoted to the study of the maximum of the excursion of a random walk with negative drift and light-tailed increments. More precisely, we determine the local asymptotics of the joint distribution of the length, maximum and the time at which this maximum is achieved. This result allows one to obtain a local central limit theorems for the length of the excursion conditioned on large values of the maximum.