Cookies

We use cookies to ensure that we give you the best experience on our website. You can change your cookie settings at any time. Otherwise, we'll assume you're OK to continue.

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

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 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) .