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

Georgiou, Nicholas; Menshikov, Mikhail V.; Mijatovic, Aleksandar; Wade, Andrew R.

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Authors

Aleksandar Mijatovic



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.

Citation

Georgiou, N., Menshikov, M. V., Mijatovic, A., & Wade, A. R. (2016). Anomalous recurrence properties of many-dimensional zero-drift random walks. Advances in Applied Probability, 48(Issue A), 99-118. https://doi.org/10.1017/apr.2016.44

Journal Article Type Article
Acceptance Date Sep 30, 2015
Online Publication Date Jul 25, 2016
Publication Date Jul 25, 2016
Deposit Date Apr 20, 2016
Publicly Available Date Apr 20, 2016
Journal Advances in Applied Probability
Print ISSN 0001-8678
Electronic ISSN 1475-6064
Publisher Applied Probability Trust
Peer Reviewed Peer Reviewed
Volume 48
Issue Issue A
Pages 99-118
DOI https://doi.org/10.1017/apr.2016.44

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Copyright Statement
© Copyright Applied Probability Trust 2016.




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