  Sequential Monte Carlo (SMC) methods, also known as particle filters,
constitute a class of algorithms used to approximate expectations with respect
to a sequence of probability distributions as well as the normalising constants
of those distributions. Sequential MCMC methods are an alternative class of
techniques addressing similar problems in which particles are sampled according
to an MCMC kernel rather than conditionally independently at each time step.
These methods were introduced over twenty years ago by Berzuini et al. (1997).
Recently, there has been a renewed interest in such algorithms as they
demonstrate an empirical performance superior to that of SMC methods in some
applications. We establish a strong law of large numbers and a central limit
theorem for sequential MCMC methods and provide conditions under which errors
can be controlled uniformly in time. In the context of state-space models, we
provide conditions under which sequential MCMC methods can indeed outperform
standard SMC methods in terms of asymptotic variance of the corresponding Monte
Carlo estimators.
