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

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).