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 N≧2. 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<p<e implies ET 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=∞.
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This paper constitutes a portion of the author's Ph.D. dissertation written at the University of Illinois
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McConnell, T.R. Exit times of N-dimensional random walks. Z. Wahrscheinlichkeitstheorie verw Gebiete 67, 213–233 (1984). https://doi.org/10.1007/BF00535269
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DOI: https://doi.org/10.1007/BF00535269