Scaling limit for local times and return times of a randomly biased walk on a Galton-Watson tree

TL;DR

This paper studies the asymptotic behavior of local times and return times of a biased random walk on a Galton-Watson tree, showing convergence to stable Lévy processes, extending previous results by Hu.

Contribution

It extends existing results by analyzing the scaling limits of local and return times for a biased walk on Galton-Watson trees, revealing convergence to stable Lévy processes.

Findings

Local time at the root converges to a maximum of a stable Lévy process.

Return times to the root converge to a hitting time of a stable Lévy process.

Results generalize previous work by Hu (2017).

Abstract

We consider a null recurrent random walk $\mathbb{X}$ on a super-critical Galton Watson marked tree $\mathbb{T}$ in the (sub-)diffusive regime. We are interested in the asymptotic behaviour of the local time of its root at $n$, which is the total amount of time spent by the random walk $\mathbb{X}$ on the root of $\mathbb{T}$ up to the time $n$, and in its $n$-th return time to the root of $\mathbb{T}$. We show that properly renormalized, this local time and this $n$-th return time respectively converge in law to the maximum and to an hitting time of some stable L\'evy process. This paper aims in particular to extent the results of Y. Hu [Hu17].