The Annals of Applied Probability

Ergodicity of the zigzag process

Joris Bierkens, Gareth O. Roberts, and Pierre-André Zitt

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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 (Markov Chains and Stochastic Stability (2009) Cambridge Univ. Press; Adv. in Appl. Probab. 25 (1993) 487–517). 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.

Article information

Ann. Appl. Probab., Volume 29, Number 4 (2019), 2266-2301.

Received: December 2017
Revised: October 2018
First available in Project Euclid: 23 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 65C05: Monte Carlo methods

Piecewise deterministic Markov process irreducibility ergodicity exponential ergodicity central limit theorem


Bierkens, Joris; Roberts, Gareth O.; Zitt, Pierre-André. Ergodicity of the zigzag process. Ann. Appl. Probab. 29 (2019), no. 4, 2266--2301. doi:10.1214/18-AAP1453.

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