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Random walks in a quarter plane with zero drifts: transience and recurrence

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

I. M. Asymont ,
G. Fayolle  and
Μ. V. Menshikov
I. M. Asymont*
Affiliation:
Moscow State University
G. Fayolle*
Affiliation:
INRIA
Μ. V. Menshikov*
Affiliation:
Moscow State University
*
∗Postal address: Moscow State University, Mechanico-Mathematical Faculty, Chair of Probability, Laboratory of Large Random Systems, Vorobyevy Gori, 119899 Moscow, Russia.
∗∗Postal address: INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France.
∗Postal address: Moscow State University, Mechanico-Mathematical Faculty, Chair of Probability, Laboratory of Large Random Systems, Vorobyevy Gori, 119899 Moscow, Russia.
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Abstract

In this paper we continue the study of the classification problem for random walks in the quarter plane, with zero drifts in the interior of the domain. The necessary and sufficient conditions for these random walks to be ergodic were found earlier in [3]. Here we obtain necessary and sufficient conditions for transience by constructing suitable Lyapounov functions.

Keywords

LYAPOUNOV FUNCTIONTRANSIENCE CONDITIONS

MSC classification

Secondary: 60G42: Martingales with discrete parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 4 , December 1995 , pp. 941 - 955
Copyright
Copyright © Applied Probability Trust 1995 

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References

[1] Fayolle, G. (1989) On random walks arising in queueing systems: Ergodicity and transience via quadratic form as Lapounov functions — Part I. Queueing Systems 5, 167–184.Google Scholar
[2] Fayolle, G. and Iasnogorodski, I. (1987) Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol. J. Appl. Prob. 24, 347–354.Google Scholar
[3] Fayolle, G., Malyshev, V. A. and Menshikov, M. V. (1992) Random walks in a quarter plane with zero drifts. I. Ergodicity and null recurrence. Ann. Inst. H. Poincaré Probabilités et Statistiques 28(2) 179–194.Google Scholar
[4] Karlin, S. (1968) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
[5] Maylshev, V. A. (1972) Classification of two-dimensional positive random walks and almost linear semimartingales. Dokl. Akad. Nauk. SSSR, 202, 526–528. English translation in Soviet Math. Dokl. 13 (1972) Google Scholar
[6] Mertens, J. F., Samuel-Cahn, E. and Zamir, S. (1978) Necessary and sufficient conditions for recurrence of Markov chains in terms of inequalities. J. Appl. Prob. 15, 848–851.Google Scholar
[7] Williams, R. (1985) Recurrence classification and invariant measure for reflected Brownian motion in a wedge. Ann. Prob. 13, 758–778.Google Scholar