Discrete-time risk-sensitive filters with non-Gaussian initial conditions and their ergodic properties

We study asymptotic stability properties of risk-sensitive filters with respect to their initial conditions. In particular, we consider a linear time-invariant systems with initial conditions that are not necessarily Gaussian. We show that in the case of Gaussian initial conditions, the optimal risk-sensitive filter asymptotically converges to any suboptimal filter initialized with an incorrect covariance matrix for the initial state vector in the mean square sense, provided the incorrect initializing value for the covariance matrix results in a risk-sensitive filter that is asymptotically stable. For non-Gaussian initial conditions, we show that under certain conditions, a suboptimal risk-sensitive filter initialized with Gaussian initial conditions asymptotically approaches the optimal risk-sensitive filter for non-Gaussian initial conditions in the mean square sense.