The Several Dimensional Gambler's Ruin Problem

TL;DR

This paper analyzes the asymptotic behavior of exit times and maxima for multidimensional random walks in the gambler's ruin problem, providing new limit expressions and simplified proofs using optional stopping and Brownian scaling.

Contribution

It introduces multidimensional refinements of Erdős-Kac type theorems for gambler's ruin, offering simplified proofs and new asymptotic limit expressions for exit times and maxima.

Findings

Asymptotic expressions for all p-moments of exit times from L-infinity balls.

Asymptotic expressions for p-moments of partial maxima in the same metric.

Simplified proof approach using optional stopping theorem and Brownian motion scaling.

Abstract

We consider the simple random walk on the $N$-dimensional integer lattice from the perspective of evaluating asymptotically the duration of play in the multidimensional gambler\apost s ruin problem. We show that, under suitable rescalings, all $p$-moments of exit-times from balls in the $L$-infinity metric, and all $p$-moments of partial-maxima values in this metric, possess associated asymptotic limit expressions, admitting two representations each. We derive for this purpose multidimensional refinements of the corresponding two-folded extension of Erd\H os-Kac theorem, which we revisit to this end. We show in particular a simplifying proof approach, which relies on an application of the optional stopping theorem, and yields the corresponding first-passage times asymptotics in parallel. We observe a direct manner of proof of the relation among the two limit expressions by Brownian motion scaling. We indicate in a manner intended to be brief and comprehensive other known proof approaches for the purposes of comparison and completeness.