Heavy-tailed random walks on complexes of half-lines

TL;DR

This paper analyzes the recurrence behavior of heavy-tailed random walks on a complex of half-lines connected at a point, providing a cotangent-based criterion and studying moments of return times, extending classical oscillating random walk results.

Contribution

It introduces a general cotangent criterion for recurrence of heavy-tailed walks on complexes of half-lines, including new moment existence results and non-linearity insights.

Findings

Recurrence classification depends on the sign of a cotangent sum involving tail exponents and stationary distribution.

The model generalizes oscillating random walks and reveals non-linear recurrence criteria.

Sharp results on the existence of polynomial moments of return times, including new cases for symmetric heavy-tailed walks.

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