The Distribution of Heights of Discrete Excursions

TL;DR

This paper derives the limiting distribution of the height of a class of discrete excursions, showing it converges to the supremum of a Brownian excursion, extending known results for Dyck and Motzkin paths.

Contribution

It introduces a new analytical approach using Schur polynomial representations and Mellin transforms to analyze the height distribution of discrete excursions.

Findings

Limiting distribution is the supremum of a Brownian excursion.

Applicable to excursions with one positive step and finite non-positive steps.

Extends known results from Dyck and Motzkin paths.

Abstract

We compute the limiting distribution of height of a random discrete excursion with step sets consisting of one positive step 1 and arbitrary finite set of non-positive integers. The limit law is the supremum of a Brownian excursion. This is well-known for Dyck and Motzkin paths. We apply a representation of the length and height generating function in terms of certain Schur polynomials put forward in a 2008 paper by Bousquet-Melout which leads to a form of the moment generating functions amenable to a Mellin transform analysis.