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Overshoots over curved boundaries. II

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

R. A. Doney  and
P. S. Griffin
R. A. Doney*
Affiliation:
University of Manchester
P. S. Griffin*
Affiliation:
Syracuse University
*
∗ Postal address: Mathematics Department, University of Manchester, 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 continue the study of the asymptotic behaviour of a random walk when it exits from a symmetric region of the form {(x, n): |x| ≤ rnb} as r → ∞ which was begun in Part I of this work. In contrast to that paper, we are interested in the case where the probability of exiting at the upper boundary tends to 1. In this scenario we treat the case where the power b lies in the interval [0, 1), and we establish necessary and sufficient conditions for the overshoot to be relatively stable in probability (except for the case ), and for the pth moment of the overshoot to be O(rq) as r → ∞.

Keywords

Random walkexit timepower-law boundaryrelative stabilitymoments of overshoots

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 4 , December 2004 , pp. 1148 - 1174
Copyright
Copyright © Applied Probability Trust 2004 

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Footnotes

Supported by EPSRC grant GR/N 94939.

References

[1] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press.Google Scholar
[2] Doney, R. A. (1997). One-sided local large deviation and renewal theorems in the case of infinite mean. Prob. Theory Relat. Fields 107, 451–465.CrossRefGoogle Scholar
[3] Doney, R. A. and Griffin, P. S. (2003). Overshoots over curved boundaries. Adv. Appl. Prob. 35, 417–448.CrossRefGoogle Scholar
[4] Doney, R. A. and Maller, R. A. (2000). Random walks crossing curved boundaries: a functional limit theorem, stability and asymptotic distributions for exit times and positions. Adv. Appl. Prob. 32, 1117–1149.CrossRefGoogle Scholar
[5] Esseen, C. G. (1968). On the concentration function of a sum of independent random variables. Z. Wahrscheinlichkeitsth. 9, 290–308.CrossRefGoogle Scholar
[6] Griffin, P. S. and Maller, R. A. (1998). On the rate of growth of the overshoot and the maximum partial sum. Adv. Appl. Prob. 30, 181–196.CrossRefGoogle Scholar
[7] Griffin, P. S. and McConnell, T. R. (1992). On the position of a random walk at the time of first exit from a sphere. Ann. Prob. 20, 825–854.CrossRefGoogle Scholar
[8] Griffin, P. S. and McConnell, T. R. (1995). Lp-boundedness of the overshoot in multidimensional renewal theory. Ann. Prob. 23, 2022–2056.CrossRefGoogle Scholar
[9] Kesten, H. and Maller, R. A. (1996). Two renewal theorems for general random walks tending to infinity. Prob. Theory Relat. Fields 106, 1–38.CrossRefGoogle Scholar
[10] Pruitt, W. E. (1981). The growth of random walks and Lévy processes. Ann. Prob. 9, 948–956.CrossRefGoogle Scholar
[11] Stout, W. F. (1974). Almost Sure Convergence (Prob. Math. Statist. 24). Academic Press, New York.Google Scholar