GAMBLERS RUIN AND THE FIRST EXIT POSITION OF RANDOM-WALK FROM LARGE SPHERES

Let T-r be the first time a sum S-n of nondegenerate i.i.d. random vectors in R(d) leaves the sphere of radius r in some given norm. We characterize, in terms of the distribution of the individual summands, the following probabilistic behavior: S-Tr/parallel to S(Tr)parallel to has no subsequential weak limit supported on a closed half-space. In one dimension, this result solves a very general form of the gambler's ruin problem. We also characterize the existence of degenerate limits and obtain analogous results for triangular arrays along any subsequence r(k) --> infinity. Finally, we compute the limiting joint distribution of (parallel to S(Tr)parallel to - r, S-Tr/parallel to S(Tr)parallel to).