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 ${\mu }_k$. If ${\chi }_k$ is $1$ for one-sided half-lines $k$ and $1/2$ for two-sided half-lines, and ${\alpha }_k$ is the tail exponent of the jumps on half-line $k$, we show that the recurrence classification for the case where all ${\alpha }_k{\chi }_k\in (0,1)$ is determined by the sign of ${\sum }_k{\mu }_k\cot ({\chi }_k\pi {\alpha }_k)$. 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 ${\alpha }_1$ and ${\alpha }_2$; 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 $\alpha \in (1,2)$.

35 pages, 2 figures

References in corpus (1)

• Recurrence and transience criteria for two cases of stable-like Markov chains

Cited by in corpus (1)

• Markov chains with heavy-tailed increments and asymptotically zero drift