Ergodicity of the zigzag process

TL;DR

This paper proves the ergodicity and convergence properties of the zigzag process, a piecewise deterministic Markov process used in MCMC sampling, under weak assumptions, and establishes a central limit theorem for empirical averages.

Contribution

It provides the first rigorous proof of ergodicity for the zigzag process under minimal conditions, using the Meyn-Tweedie approach.

Findings

Proves convergence of the zigzag process to its target distribution.

Establishes a central limit theorem for empirical averages.

Shows the process can reach all points in the space even with minimal switching rates.

Abstract

The zigzag process is a Piecewise Deterministic Markov Process which can be used in a MCMC framework to sample from a given target distribution. We prove the convergence of this process to its target under very weak assumptions, and establish a central limit theorem for empirical averages under stronger assumptions on the decay of the target measure. We use the classical "Meyn-Tweedie" approach. The main difficulty turns out to be the proof that the process can indeed reach all the points in the space, even if we consider the minimal switching rates.