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

Georgiou, Nicholas and Menshikov, Mikhail V. and Mijatovic, Aleksandar and Wade, Andrew R. (2016) 'Anomalous recurrence properties of many-dimensional zero-drift random walks.', Advances in applied probability., 48 (Issue A). pp. 99-118.


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.

Item Type:Article
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Publisher statement:© Copyright Applied Probability Trust 2016.
Date accepted:30 September 2015
Date deposited:20 April 2016
Date of first online publication:25 July 2016
Date first made open access:No date available

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