  We consider the problem of approximating optimal in the Minimum Mean Squared
Error (MMSE) sense nonlinear filters in a discrete time setting, exploiting
properties of stochastically convergent state process approximations. More
specifically, we consider a class of nonlinear, partially observable stochastic
systems, comprised by a (possibly nonstationary) hidden stochastic process (the
state), observed through another conditionally Gaussian stochastic process (the
observations). Under general assumptions, we show that, given an approximating
process which, for each time step, is stochastically convergent to the state
process, an approximate filtering operator can be defined, which converges to
the true optimal nonlinear filter of the state in a strong and well defined
sense. In particular, the convergence is compact in time and uniform in a
completely characterized measurable set of probability measure almost unity,
also providing a purely quantitative justification of Egoroff's Theorem for the
problem at hand. The results presented in this paper can form a common basis
for the analysis and characterization of a number of heuristic approaches for
approximating optimal nonlinear filters, such as approximate grid based
techniques, known to perform well in a variety of applications.
