  Sequential Monte Carlo (SMC) methods, also known as particle filters, are
simulation-based recursive algorithms for the approximation of the a posteriori
probability measures generated by state-space dynamical models. At any given
time $t$, a SMC method produces a set of samples over the state space of the
system of interest (often termed "particles") that is used to build a discrete
and random approximation of the posterior probability distribution of the state
variables, conditional on a sequence of available observations. One potential
application of the methodology is the estimation of the densities associated to
the sequence of a posteriori distributions. While practitioners have rather
freely applied such density approximations in the past, the issue has received
less attention from a theoretical perspective. In this paper, we address the
problem of constructing kernel-based estimates of the posterior probability
density function and its derivatives, and obtain asymptotic convergence results
for the estimation errors. In particular, we find convergence rates for the
approximation errors that hold uniformly on the state space and guarantee that
the error vanishes almost surely as the number of particles in the filter
grows. Based on this uniform convergence result, we first show how to build
continuous measures that converge almost surely (with known rate) toward the
posterior measure and then address a few applications. The latter include
maximum a posteriori estimation of the system state using the approximate
derivatives of the posterior density and the approximation of functionals of
it, for example, Shannon's entropy.
  This manuscript is identical to the published paper, including a gap in the
proof of Theorem 4.2. The Theorem itself is correct. We provide an {\em
erratum} at the end of this document with a complete proof and a brief
discussion.
