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

2014/01/25 by Crisan, Dan, Li, Kai
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1401.6592

Abstract

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/√(n). This implies, in particular, that the mean square error of the approximation as defined in [4, 5] is of order 1/n.

Related

• Efficient Gaussian Mixture Filters based on Transition Density Approximation
• Gaussian filters for nonlinear filtering problems
• Generalised Particle Filters with Gaussian Mixtures
• On the best approximation by finite Gaussian mixtures
• Wiener Filters in Gaussian Mixture Signal Estimation with Infinity-Norm Error