On the nice behaviour of the Gaussian projection filter with small observation noise

When projecting on the manifold of Gaussian densities, the projection filter has been shown to be equal to a McShane-Fisk-Stratonovich (MFS) derivation of the Gaussian assumed density filter. Starting from this point, we study the asymptotic behaviour of the Gaussian projection filter when the covariance of the observation noise tends to zero. We prove that the mean square difference between the true state of the system and the estimate given by the projection filter is bounded by a constant which is proportional to the magnitude of the observation noise.