  We study the forgetting properties of the particle filter when its state -
the collection of particles - is regarded as a Markov chain. Under a strong
mixing assumption on the particle filter's underlying Feynman-Kac model, we
find that the particle filter is exponentially mixing, and forgets its initial
state in $O(\log N )$ `time', where $N$ is the number of particles and time
refers to the number of particle filter algorithm steps, each comprising a
selection (or resampling) and mutation (or prediction) operation. We present an
example which suggests that this rate is optimal. In contrast to our result,
available results to-date are extremely conservative, suggesting $O(\alpha^N)$
time steps are needed, for some $\alpha>1$, for the particle filter to forget
its initialisation. We also study the conditional particle filter (CPF) and
extend our forgetting result to this context. We establish a similar
conclusion, namely, CPF is exponentially mixing and forgets its initial state
in $O(\log N )$ time. To support this analysis, we establish new time-uniform
$L^p$ error estimates for CPF, which can be of independent interest.
