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. 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
