  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.
