A central limit type theorem for Gaussian mixture approximations to the nonlinear filtering problem

Approximating the solution of the nonlinear filtering problem with Gaussian mixtures has been a very popular method since the 1970s. However, the vast majority of such approximations are introduced in an ad-hoc manner without theoretical grounding. This work is a continuation of [4, 5], where we described a rigorous Gaussian mixture approximation to the solution of the filtering problem. We deduce here a refined estimate of the rate of convergence of the approximation. We do this by proving a central limit type theorem for the error process. We also find the optimal variances of the Gaussian measures are of order $1/\sqrt{n}$. This implies, in particular, that the mean square error of the approximation as defined in [4, 5] is of order $1/n$.