Exit times of N-dimensional random walks

Summary We study the exit time T of a sum of independent, identically distributed random vectors, X ₁, X ₂, ..., from a subset R of N dimensional Euclidean space when N2. We assume that R is invariant under positive dilations and that the boundary of R satisfies certain regularity conditions. The random vector X ₁ is to have mean zero and a nonsingular covariance matrix. We show that there is a critical exponent, e, independent of X ₁, X ₂, ..., such that 0<pET ^p/2<. In addition, if X ₁ is bounded and slightly more restrictive assumptions are imposed on R, then p>e implies ET ^p/2=.This paper constitutes a portion of the author's Ph.D. dissertation written at the University of Illinois