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Durham University

Department of Mathematical Sciences

Academic Staff

Publication details for Mikhail Menshikov

Menshikov, Mikhail V., Petritis, Dimitri & Wade, Andrew R. (2018). Heavy-tailed random walks on complexes of half-lines. Journal of Theoretical Probability 31(3): 1819-1859.

Author(s) from Durham


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 μkμk . If χkχk is 1 for one-sided half-lines k and 1 / 2 for two-sided half-lines, and αkαk 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)αkχk∈(0,1) is determined by the sign of ∑kμkcot(χkπαk)∑kμkcot⁡(χkπαk) . In the case of two half-lines, the model fits naturally on RR and is a version of the oscillating random walk of Kemperman. In that case, the cotangent criterion for recurrence becomes linear in α1α1 and α2α2 ; 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 RR with symmetric increments of tail exponent α∈(1,2)α∈(1,2) .