Limit theorems for the zig-zag process

Joris Bierkens, Andrew Duncan

Research output: Contribution to JournalArticleAcademicpeer-review


Markov chain Monte Carlo (MCMC) methods provide an essential tool in statistics for sampling from complex probability distributions. While the standard approach to MCMC involves constructing discrete-time reversible Markov chains whose transition kernel is obtained via the Metropolis-Hastings algorithm, there has been recent interest in alternative schemes based on piecewise deterministic Markov processes (PDMPs). One such approach is based on the zig-zag process, introduced in Bierkens and Roberts (2016), which proved to provide a highly scalable sampling scheme for sampling in the big data regime; see Bierkens et al. (2016). In this paper we study the performance of the zig-zag sampler, focusing on the one-dimensional case. In particular, we identify conditions under which a central limit theorem holds and characterise the asymptotic variance. Moreover, we study the influence of the switching rate on the diffusivity of the zig-zag process by identifying a diffusion limit as the switching rate tends to. Based on our results we compare the performance of the zig-zag sampler to existing Monte Carlo methods, both analytically and through simulations.

Original languageEnglish
Pages (from-to)791-825
Number of pages35
JournalAdvances in Applied Probability
Issue number3
Publication statusPublished - 1 Sep 2017
Externally publishedYes


  • central limit theorem
  • continuous-time Markov process
  • functional central limit theorem
  • MCMC
  • nonreversible Markov process
  • piecewise deterministic Markov process


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