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 $μ_k$. If $χ_k$ is $1$ for one-sided half-lines $k$ and $1/2$ for two-sided half-lines, and $α_k$ is the tail exponent of the jumps on half-line $k$, we show that the recurrence classification for the case where all $α_k χ_k \in (0,1)$ is determined by the sign of $\sum_k μ_k \cot ( χ_k πα_k )$. In the case of two half-lines, the model fits naturally on $\mathbb{R}$ and is a version of the oscillating random walk of Kemperman. In that case, the cotangent criterion for recurrence becomes linear in $α_1$ and $α_2$; our general setting exhibits the essential non-linearity 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 $\mathbb{R}$ with symmetric increments of tail exponent $α\in (1,2)$.