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Limit theorems for the zig-zag process

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations Limit theorems

Published online by Cambridge University Press:  08 September 2017

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

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.

Keywords

MCMCnonreversible Markov processpiecewise deterministic Markov processcontinuous-time Markov processcentral limit theoremfunctional central limit theorem

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 3 , September 2017 , pp. 791 - 825
Copyright
Copyright © Applied Probability Trust 2017 

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