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Anomalous recurrence properties of many-dimensional zero-drift random walks

Part of:Markov processesStochastic processes

Published online by Cambridge University Press:  25 July 2016

Nicholas Georgiou,
Mikhail V. Menshikov,
Aleksandar Mijatović and
Andrew R. Wade
Nicholas Georgiou*
Affiliation:
Durham University and Heilbronn Institute for Mathematical Research
Mikhail V. Menshikov*
Affiliation:
Durham University
Aleksandar Mijatović*
Affiliation:
King's College London
Andrew R. Wade*
Affiliation:
Durham University
*
Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. Email address: nicholas.georgiou@durham.ac.uk
Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. Email address: mikhail.menshikov@durham.ac.uk
Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK. Email address: aleksandar.mijatovic@kcl.ac.uk
Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. Email address: andrew.wade@durham.ac.uk
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Abstract

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Famously, a d-dimensional, spatially homogeneous random walk whose increments are nondegenerate, have finite second moments, and have zero mean is recurrent if d∈{1,2}, but transient if d≥3. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially nonhomogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension d≥2, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by random walks whose increments are supported on ellipsoids that are symmetric about the ray from the origin through the walk's current position; these elliptic random walks generalize the classical homogeneous Pearson‒Rayleigh walk (the spherical case). Our proof of the recurrence classification is based on fundamental work of Lamperti.

Keywords

Nonhomogeneous random walkelliptic random walkzero driftrecurrencetransience

MSC classification

Primary:60J05: Discrete-time Markov processes on general state spaces
Secondary:60J10: Markov chains (discrete-time Markov processes on discrete state spaces)60G42: Martingales with discrete parameter60G50: Sums of independent random variables; random walks

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue A: Probability, Analysis and Number Theory , July 2016, pp. 99 - 118
Copyright
Copyright © Applied Probability Trust 2016 

References

[1] Berg, H. C. (1993).Random Walks in Biology, expanded edn.Princeton University Press.Google Scholar
[2] Bingham, N. H. (1972).Random walk on spheres.Z. Wahrscheinlichkeitsth. 22,169–192.CrossRefGoogle Scholar
[3] Carazza, B. (1977).The history of the random-walk problem: considerations on the interdisciplinarity in modern physics.Rivista del Nuovo Cimento Serie 2 7,419–427.CrossRefGoogle Scholar
[4] Chung, K. L. and Fuchs, W. H. J. (1951).On the distribution of values of sums of random variables.Mem. Amer. Math. Soc. 6, 12pp.Google Scholar
[5] Gut, A. (2005).Probability: A Graduate Course(Springer Texts Statist. 75).Springer,New York.Google Scholar
[6] Hryniv, O.,Menshikov, M. V. and Wade, A. R. (2013).Excursions and path functionals for stochastic processes with asymptotically zero drifts.Stoch. Process. Appl. 123,1891–1921.CrossRefGoogle Scholar
[7] Hryniv, O.,MacPhee, I. M.,Menshikov, M. V. and Wade, A. R. (2012).Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips.Electron. J. Prob. 17, article 59, 28pp.CrossRefGoogle Scholar
[8] Hughes, B. D. (1995).Random Walks and Random Environments, Vol. 1.Oxford University Press.CrossRefGoogle Scholar
[9] Kallenberg, O. (2002).Foundations of Modern Probability,2nd edn.Springer,New York.CrossRefGoogle Scholar
[10] Kemperman, J. H. B. (1974).The oscillating random walk.Stoch. Process. Appl. 2,1–29.CrossRefGoogle Scholar
[11] Kingman, J. F. C. (1963).Random walks with spherical symmetry.Acta Math. 109,11–53.CrossRefGoogle Scholar
[12] Lamperti, J. (1960).Criteria for the recurrence or transience of stochastic process, I.J. Math. Anal. Appl. 1,314–330.CrossRefGoogle Scholar
[13] Menshikov, M. V.,Vachkovskaia, M. and Wade, A. R. (2008).Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains.J. Statist. Phys. 132,1097–1133.CrossRefGoogle Scholar
[14] Nossal, R. (1983).Stochastic aspects of biological locomotion.J. Statist. Phys. 30,391–400.CrossRefGoogle Scholar
[15] Nossal, R. J. and Weiss, G. H. (1974).A generalized Pearson random walk allowing for bias.J. Statist. Phys. 10,245–253.CrossRefGoogle Scholar
[16] Pearson, K. and Blakeman, J. (1906).A Mathematical Theory of Random Migration (Drapers' Company Research Memoirs, Biometric Series 3).Dulau and Co.,London.CrossRefGoogle Scholar
[17] Pearson, K. and Lord Rayleigh, (1905).The problem of the random walk.Nature 72, pp. 294, 318, 342.CrossRefGoogle Scholar
[18] Peres, Y.,Popov, S. and Sousi, P. (2013).On recurrence and transience of self-interacting random walks.Bull. Brazilian Math. Soc. (N.S.) 44,841–867.CrossRefGoogle Scholar
[19] Rogozin, B. A. and Foss, S. G. (1978).Recurrency of an oscillating random walk.Theory Prob. Appl. 23,155–162. Translated from Teor. Veroyatn. Primen. 23 (1978)161–169.CrossRefGoogle Scholar
[20] Zeitouni, O. (2006).Lecture Notes on Random Walks in Random Environment. 9th May 2006 version. Available at http://www.wisdom.weizmann.ac.il/~zeitouni/.Google Scholar