Limit theorems and structural properties of the Cat-and-Mouse Markov chain

TL;DR

This paper analyzes the Cat-and-Mouse Markov chain, extending previous models by considering more general jump distributions and higher dimensions, and derives new limit theorems and structural properties.

Contribution

It introduces generalized jump distributions for the model and explores its behavior in higher dimensions, providing new limit laws and structural insights.

Findings

Scaling limit for the second component in 2D with general jumps

Limiting law for the last component in higher dimensions

Structural properties of multi-dimensional chains

Abstract

We revisit the so-called Cat-and-Mouse Markov chain, studied earlier by Litvak and Robert (2012). This is a 2-dimensional Markov chain on the lattice $\mathbb{Z}^2$, where the first component (the cat) is a simple random walk and the second component (the mouse) changes when the components meet. We obtain new results for two generalisations of the model. Firstly, in the 2-dimensional case we consider far more general jump distributions for the components and obtain a scaling limit for the second component. When we let the first component be a simple random walk again, we further generalise the jump distribution of the second component. Secondly, we consider chains of three and more dimensions, where we investigate structural properties of the model and find a limiting law for the last component.