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

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, the maximum, and the time at which this maximum is achieved. This result allows one to obtain a local central limit theorem for the length of the excursion conditioned on large values of the maximum.