  Markov chain Monte Carlo 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 (Bierkens,
Fearnhead and Roberts (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 (CLT) holds and
characterize 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 infinity. Based on our results
we compare the performance of the Zig-Zag sampler to existing Monte Carlo
methods, both analytically and through simulations.
