Fluctuations of random Motzkin paths

TL;DR

This paper investigates the joint fluctuations of counting processes in random Motzkin paths, showing they converge to combinations of Brownian motion and Brownian excursion, extending understanding of their probabilistic structure.

Contribution

It introduces a joint convergence result for counting processes in Motzkin paths, linking them to classical stochastic processes, with novel integral representations and Markov process analysis.

Findings

Counting processes converge jointly to Brownian motion and Brownian excursion

Fluctuations are described by linear combinations of these processes

The approach uses Laplace transforms and integral identities

Abstract

It is known that after scaling a random Motzkin path converges to a Brownian excursion. We prove that the fluctuations of the counting processes of the ascent steps, the descent steps and the level steps converge jointly to linear combinations of two independent processes: a Brownian motion and a Brownian excursion. The proofs rely on the Laplace transforms and an integral representation based on an identity connecting non-crossing pair partitions and joint moments of an explicit non-homogeneous Markov process.