Heavy-tailed random walks on complexes of half-lines

Mikhail V. Menshikov, Dimitri Petritis, and Andrew R. Wade

Journal of Theoretical Probability 31, no. 3, September 2018, 1819–1859.

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 \emph{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 $\alpha \in (1,2)$.