  For particle filters and ensemble Kalman filters it is of practical
importance to understand how and why data assimilation methods can be effective
when used with a fixed small number of particles, since for many large-scale
applications it is not practical to deploy algorithms close to the large
particle limit asymptotic. In this paper we address this question for particle
filters and, in particular, study their accuracy (in the small noise limit) and
ergodicity (for noisy signal and observation) without appealing to the large
particle number limit. We first overview the accuracy and minorization
properties for the true filtering distribution, working in the setting of
conditional Gaussianity for the dynamics-observation model. We then show that
these properties are inherited by optimal particle filters for any fixed number
of particles, and use the minorization to establish ergodicity of the filters.
For completeness we also prove large particle number consistency results for
the optimal particle filters, by writing the update equations for the
underlying distributions as recursions. In addition to looking at the optimal
particle filter with standard resampling, we derive all the above results for
(what we term) the Gaussianized optimal particle filter and show that the
theoretical properties are favorable for this method, when compared to the
standard optimal particle filter.
