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A note on conditioned random walk

Published online by Cambridge University Press:  14 July 2016

R. A. Doney
R. A. Doney*
Affiliation:
University of Manchester
*
∗ Postal address: Statistical Laboratory, Department of Mathematics, The University, Manchester M13 9PL, U.K.
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Abstract

This note is concerned with N, the time at which a random walk first exits from [0,∞), and M, the maximum of the random walk up to time N. In the case that the random walk has zero mean and finite variance, simple proofs are given of asymptotic estimates for P{M > x}, P{N ≦ ux2|M > x} and P{M ≦ v √n|N > n}.

Keywords

RANDOM WALKMAXIMUMLADDER VARIABLESCONDITIONED LIMIT THEOREMS
Type
Short Communications
Information
Journal of Applied Probability , Volume 20 , Issue 2 , June 1983 , pp. 409 - 412
Copyright
Copyright © Applied Probability Trust 1983 

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References

[1] Bolthausen, E. (1976) On a functional central limit theorem for random walks conditioned to stay positive. Ann. Prob. 4, 480–485.CrossRefGoogle Scholar
[2] Durrett, R. T. and Iglehart, O. L. (1977) Functionals of Brownian meander. Ann. Prob. 5, 130–135.CrossRefGoogle Scholar
[3] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn. Wiley, New York.Google Scholar
[4] Lindvall, T. (1976) On the maximum of a branching process. Scand. J. Statist. 3, 209–214.Google Scholar
[5] Pakes, A. G. (1978) On the maximum and absorption time of left-continuous random walk. J. Appl. Prob. 15, 292–299.Google Scholar
[6] Rogozin, B. A. (1972) The distribution of the first hit for stable and asymptotically stable walks on an interval. Theory Prob. Appl. 17, 332–338.Google Scholar
[7] Shimura, M. (1981) The maximum and time to absorption of a recurrent random walk with finite second moment.Google Scholar
[8] Shimura, M. (1981) A class of conditional limit theorems related to ruin problem.Google Scholar