A bound for the error covariance of the recursive Kalman filter with Markov jump parameters

In this paper we study the error covariance matrix of the recursive Kalman filter when the parameters of the filter are driven by a Markov chain taking values in a countably infinite set. In this context, the error covariance matrix of the filter depends on the Markov state realizations, and in this sense forms a stochastic process. We show in a rather direct and comprehensive manner that a standard stochastic detectability concept plays the role of a sufficient condition for the mean of the error covariance process of the Kalman filter to be bounded. Illustrative examples are included.