The Several Dimensional Gambler's Ruin Problem

摘要

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'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 Erdos-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.