Ordered exponential random walks

TL;DR

This paper analyzes a $d$-dimensional exponential random walk conditioned to stay ordered, explicitly constructs harmonic functions for the process, and explores its connections to queueing theory and asymptotic behaviors.

Contribution

It provides an explicit harmonic function for the ordered exponential random walk and constructs the process via Doob's $h$-transform, extending understanding beyond nearest-neighbor cases.

Findings

Explicit harmonic function for the ordered process

Asymptotic tail probabilities for disorder times

Fredholm determinant formulas for extremal particle distributions

Abstract

We study a $d$-dimensional random walk with exponentially distributed increments conditioned so that the components stay ordered (in the sense of Doob). We find explicitly a positive harmonic function $h$ for the killed process and then construct an ordered process using Doob's $h$-transform. Since these random walks are not nearest-neighbour, the harmonic function is not the Vandermonde determinant. The ordered process is related to the departure process of M/M/1 queues in tandem. We find asymptotics for the tail probabilities of the time until the components in exponential random walks become disordered and a local limit theorem. We find the distribution of the processes of smallest and largest particles as Fredholm determinants.