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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.

References

[1] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
[2] Doney, R. A. and Maller, R. A. (2000). Random walks crossing curved boundaries: stability and asymptotic distributions for exit times and positions. Adv. Appl. Prob. 32, 1117–1149.CrossRefGoogle Scholar
[3] 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.Google Scholar
[4] 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.Google Scholar
[5] Griffin, P. S. and McConnell, T. R. (1994). Gambler's ruin and the first exit position of a random walk from large spheres. Ann. Prob. 22, 1429–1472.Google Scholar
[6] Griffin, P. S. and McConnell, T. R. (1995). Lp-boundedness of the overshoot in multidimensional renewal theory. Ann. Prob. 23, 2022–2056.Google Scholar
[7] Gut, A. (1974). On the moments and limit distributions of some first passage times. Ann. Prob. 2, 277–308.Google Scholar
[8] Kesten, H. and Maller, R. A. (1992). Ratios of trimmed sums and order statistics. Ann. Prob. 20, 1805–1842.CrossRefGoogle Scholar
[9] Kesten, H. and Maller, R. A. (1994). Infinite limits and infinite limit points of random walks and trimmed sums. Ann. Prob. 22, 1473–1513.CrossRefGoogle Scholar
[10] Kesten, H. and Maller, R. A. (1995). The effect of trimming on the strong law of large numbers. Proc. London Math. Soc. 71, 441–480.Google Scholar
[11] Kesten, H. and Maller, R. A. (1998). Random walks crossing high level curved boundaries. J. Theoret. Prob. 11, 1019–1074.Google Scholar
[12] Kesten, H. and Maller, R. A. (1998). Random walks crossing power law boundaries. Studia Sci. Math. Hung. 34, 219–252.Google Scholar
[13] Kesten, H. and Maller, R. A. (1999). Stability and other limit laws for exit times of random walks from a strip or a halfplane. Ann. Inst. H. Poincaré Prob. Statist. 35, 685–734.Google Scholar
[14] Pruitt, W. E. (1981). The growth of random walks and Lévy processes. Ann. Prob. 9, 948–956.CrossRefGoogle Scholar