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Overshoots over curved boundaries

Published online by Cambridge University Press:  22 February 2016

R. A. Doney*
Affiliation:
Manchester University
P. S. Griffin*
Affiliation:
Syracuse University
*
Postal address: Mathematics Department, Manchester University, Manchester M13 9PL, UK. Email address: [email protected]
∗∗ Postal address: Mathematics Department, Syracuse University, Syracuse, NY 13244-1150, USA.

Abstract

We consider the asymptotic behaviour of a random walk when it exits from a symmetric region of the form {(x, n) :|x| ≤ rnb} as r → ∞. In order to be sure that this actually occurs, we treat only the case where the power b lies in the interval [0,½), and we further assume a condition that prevents the probability of exiting at either boundary tending to 0. Under these restrictions we establish necessary and sufficient conditions for the pth moment of the overshoot to be O(rq), and for the overshoot to be tight, as r → ∞.

Type
General Applied Probability
Copyright
Copyright © Applied Probability Trust 2003 

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Footnotes

Supported by EPSRC grant GR/N 94939.

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