  The particle Gibbs (PG) sampler is a systematic way of using a particle
filter within Markov chain Monte Carlo (MCMC). This results in an off-the-shelf
Markov kernel on the space of state trajectories, which can be used to simulate
from the full joint smoothing distribution for a state space model in an MCMC
scheme. We show that the PG Markov kernel is uniformly ergodic under rather
general assumptions, that we will carefully review and discuss. In particular,
we provide an explicit rate of convergence which reveals that: (i) for fixed
number of data points, the convergence rate can be made arbitrarily good by
increasing the number of particles, and (ii) under general mixing assumptions,
the convergence rate can be kept constant by increasing the number of particles
superlinearly with the number of observations. We illustrate the applicability
of our result by studying in detail two common state space models with
non-compact state spaces.
