Statistical properties of the error covariance in a Kalman filter with random measurement losses

In this paper we study statistical properties of the error covariance matrix of a Kalman filter, when it is subject to random measurement losses. We introduce a sequence of tighter upper bounds for the asymptotic expected error covariance (EEC). This sequence starts with a given upper bound in the literature and converges to the actual asymptotic EEC. Although we have not yet shown the monotonic convergence of this whole sequence, monotonic convergent subsequences are identified. The feature of these subsequences is that a tighter upper bound is guaranteed if more computation is allowed. An iterative algorithm is provided for computing each of these upper bounds. A byproduct of this paper is a more compact proof for a known necessary condition on the measurement arrival probability for the asymptotic EEC to be finite. A similar analysis leads to a necessary condition on the measurement arrival probability for the error covariance to have a finite asymptotic variance.