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Heavy-tailed random walks on complexes of half-lines.

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.


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).

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Publisher statement:© The Author(s) 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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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