  Stochastic filtering is defined as the estimation of a partially observed
dynamical system. A massive scientific and computational effort is dedicated to
the development of numerical methods for approximating the solution of the
filtering problem. Approximating the solution of the filtering problem with
Gaussian mixtures has been a very popular method since the 1970s (see
[1],[2],[46],[49]). Despite nearly fifty years of development, the existing
work is based on the success of the numerical implementation and is not
theoretically justified. This paper fills this gap and contains a rigorous
analysis of a new Gaussian mixture approximation to the solution of the
filtering problem. We deduce the L^2-convergence rate for the approximating
system and show some numerical example to test the new algorithm.
