Menshikov, Mikhail V. and Petritis, Dimitri and Wade, Andrew R. (2018) 'Heavy-tailed random walks on complexes of half-lines.', Journal of theoretical probability., 31 (3). pp. 1819-1859.
Abstract
We study a random walk on a complex of finitely many half-lines joined at a common origin; jumps are heavy-tailed and of two types, either one-sided (towards the origin) or two-sided (symmetric). Transmission between half-lines via the origin is governed by an irreducible Markov transition matrix, with associated stationary distribution μk. If χk is 1 for one-sided half-lines k and 1 / 2 for two-sided half-lines, and αk is the tail exponent of the jumps on half-line k, we show that the recurrence classification for the case where all αkχk∈(0,1) is determined by the sign of ∑kμkcot(χkπαk). In the case of two half-lines, the model fits naturally on R and is a version of the oscillating random walk of Kemperman. In that case, the cotangent criterion for recurrence becomes linear in α1 and α2; our general setting exhibits the essential nonlinearity in the cotangent criterion. For the general model, we also show existence and non-existence of polynomial moments of return times. Our moments results are sharp (and new) for several cases of the oscillating random walk; they are apparently even new for the case of a homogeneous random walk on R with symmetric increments of tail exponent α∈(1,2).
Item Type: | Article |
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Full text: | (AM) Accepted Manuscript Available under License - Creative Commons Attribution. Download PDF (430Kb) |
Full text: | (VoR) Version of Record Available under License - Creative Commons Attribution. Download PDF (Advance online version) (757Kb) |
Full text: | (VoR) Version of Record Available under License - Creative Commons Attribution. Download PDF (Final published version) (754Kb) |
Status: | Peer-reviewed |
Publisher Web site: | https://doi.org/10.1007/s10959-017-0753-5 |
Publisher statement: | © The Author(s) 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
Date accepted: | 06 March 2017 |
Date deposited: | 13 March 2017 |
Date of first online publication: | 20 March 2017 |
Date first made open access: | No date available |
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