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On the maximum and absorption time of left-continuous random walk

Published online by Cambridge University Press:  14 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
Princeton University
*
∗ On leave from the Department of Mathematics, Monash University, Clayton, Victoria, Australia.
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Abstract

In a recent paper Green (1976) obtained some conditional limit theorems for the absorption time of left-continuous random walk. His methods require that in the driftless case the increment distribution has exponentially decreasing tails and that the same is true for a transformed distribution in the case of negative drift.

Here we take a different approach which will produce Green's results under minimal conditions. Limit theorems are given for the maximum as the initial position of the random walk tends to infinity.

Keywords

LEFT-CONTINUOUS RANDOM WALKMAXIMUMABSORPTION TIMELIMIT THEOREMSLOCAL LIMIT THEOREMSRENEWAL THEOREMS

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 2 , June 1978, pp. 292 - 299
Copyright
Copyright © Applied Probability Trust 1978 

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Footnotes

Research sponsored in part by a contract with the Office of Naval Research, No. N00014–75–C–0453, awarded to the Department of Statistics, Princeton University.

References

Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Erickson, K. B. (1970) Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151, 263–291.Google Scholar
Green, P. J. (1976) The maximum and time to absorption of a left-continuous random walk. J. Appl. Prob. 13, 444–454.Google Scholar
Ibragimov, I. A. and Linnik, Yu. (1970) Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.Google Scholar
Kawata, T. (1972) Fourier Analysis in Probability Theory. Academic Press, New York.Google Scholar
Lindvall, T. (1976) On the maximum of a branching process. Scand. J. Statist. 3, 209–214.Google Scholar
Pakes, A. G. (1973) Conditional limit theorems for a left-continuous random walk. J. Appl. Prob. 10, 39–53.Google Scholar
Pakes, A. G. (1978) On the age distribution of a Markov chain. J. Appl. Prob. 15, 65–77.Google Scholar
Williamson, J. A. (1968) Random walks and Riesz kernels. Pacific J. Math. 25, 393–415.Google Scholar
Yang, Y. S. (1973) Asymptotic properties of the stationary measure of a Markov branching process. J. Appl. Prob. 10, 447–450.Google Scholar