Discrete-Time Filtering for Linear Systems with non-Gaussian Initial Conditions: Asymptotic Behavior of the Difference Between the MMSE and LMSE Estimates

We consider the one-step prediction problem for discrete-time linear systems in correlated plant and observation Gaussian white noises, with non-Gaussian initial conditions. We investigate the large time asymptotics of ϵ, the expected squared difference between the MMSE and LMSE (or Kalman) estimates of the state at time t given past observations. We characterize the limit of the error sequence {ϵ, t = 0, 1,…} and obtain some related rates of convergence; a complete analysis is provided for the scalar case. The discussion is based on explicit representations for the MMSE and LMSE estimates, recently obtained by the authors, which display the dependence of these quantities on the initial distribution.