On the convergence of the Gaussian mixture filter

This paper presents convergence results for the Box Gaussian Mixture Filter (BGMF). BGMF is a Gaussian Mixture Filter (GMF) that is based on a bank of Extended Kalman Filters. The critical part of GMF is the approximation of probability density function (pdf) as pdf of Gaussian mixture such that its components have small enough covariance matrices. Because GMF approximates prior and posterior as Gaussian mixture it is enough if we have a method to approximate arbitrary Gaussian (mixture) as a Gaussian mixture such that the components have small enough covariance matrices. In this paper, we present the Box Gaussian Mixture Approximation (BGMA) that partitions the state space into specific boxes and matches weights, means and covariances of the original Gaussian in each box to a GM approximation. If the original distribution is Gaussian mixture, BGMA does this approximation separately for each component of the Gaussian mixture. We show that BGMA converges weakly to the original Gaussian (mixture). When we apply BGMA in a Gaussian mixture filtering framework we get BGMF. We show that GMF, and also BGMF, converges weakly to the correct/exact posterior distribution.