Title: Stability of Conditional Sequential Monte Carlo,
Abstract: The particle Gibbs (PG) sampler is a Markov Chain Monte Carlo (MCMC)
algorithm, which uses an interacting particle system to perform the Gibbs
steps. Each Gibbs step consists of simulating a particle system conditioned on
one particle path. It relies on a conditional Sequential Monte Carlo (cSMC)
method to create the particle system. We propose a novel interpretation of the
cSMC algorithm as a perturbed Sequential Monte Carlo (SMC) method and apply
telescopic decompositions developed for the analysis of SMC algorithms
\cite{delmoral2004} to derive a bound for the distance between the expected
sampled path from cSMC and the target distribution of the MCMC algorithm. This
can be used to get a uniform ergodicity result. In particular, we can show that
the mixing rate of cSMC can be kept constant by increasing the number of
particles linearly with the number of observations. Based on our decomposition,
we also prove a central limit theorem for the cSMC Algorithm, which cannot be
done using the approaches in \cite{Andrieu2013} and \cite{Lindsten2014}.