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Limit theorems for the zig-zag process

Published online by Cambridge University Press:  08 September 2017

Joris Bierkens*
University of Warwick
Andrew Duncan*
Imperial College London
* Current address: Delft Institute of Applied Mathematics, Mekelweg 4, 2628 CD, Delft, The Netherlands. Email address:
** Current address: School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH, UK.


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.

Research Article
Copyright © Applied Probability Trust 2017 

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