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This technical note shows that the stationary distribution for the covariance of Kalman filtering with intermittent observations exists under mild conditions for a very general class of packet dropping models (semi-Markov chain). These results are proved using the geometric properties of Riccati recursions with respect to a particular Riemannian distance. Moreover, the Riemannian mean induced by that distance is always bounded, therefore it can be used for characterizing the performance of the system for regimes where the moments of the covariance do not exist. Other interesting properties of that mean include the symmetry between covariance and information matrices (averaging covariances or their inverse gives the same result), and its interpretation in information geometry as the “natural” mean for the manifold of Gaussian distributions.
Date of Publication: Feb. 2011