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

Part of: Markov processes Stochastic 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: [email protected]
Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. Email address: [email protected]
Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK. Email address: [email protected]
Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. Email address: [email protected]
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Abstract

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 parameter 60G50: Sums of independent random variables; random walks
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 

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