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

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

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