Local asymptotics for the area under the random walk excursion

TL;DR

This paper investigates the tail behavior of the area under a positive excursion of a negatively drifting, light-tailed random walk, providing asymptotic results and a local limit theorem for the excursion duration conditioned on large area values.

Contribution

It offers new asymptotic formulas for the probability distribution of the area and establishes a local limit theorem for excursion duration conditioned on large areas.

Findings

Derived asymptotics for tail probabilities of the area

Proved a local central limit theorem for excursion duration

Characterized the distributional behavior under large area conditioning

Abstract

We study tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and light-tailed increments. We determine the asymptotics for local probabilities for the area and prove a local central limit theorem for the duration of the excursion conditioned on the large values of its area.