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Overshoots over curved boundaries

Part of: Stochastic processes

Published online by Cambridge University Press:  22 February 2016

R. A. Doney  and
P. S. Griffin
R. A. Doney*
Affiliation:
Manchester University
P. S. Griffin*
Affiliation:
Syracuse University
*
∗ Postal address: Mathematics Department, Manchester University, Manchester M13 9PL, UK. Email address: rad@maths.man.ac.uk
∗∗ Postal address: Mathematics Department, Syracuse University, Syracuse, NY 13244-1150, USA.
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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 → ∞.

Keywords

Random walksexit timesrelative stabilitytightnessmoments of overshoots

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 417 - 448
Copyright
Copyright © Applied Probability Trust 2003 

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Footnotes

Supported by EPSRC grant GR/N 94939.

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