Discrete-time filtering for linear systems with non-Gaussian initial conditions: asymptotic behavior of the difference between the MMSE and LMSE estimates

The authors consider the one-step prediction problem for discrete-time linear systems in correlated plant and observation Gaussian white noises, with nonGaussian initial conditions. They investigate the large time asymptotics of ε_t, the expected squared difference between the MMSE and LMSE (or Kalman) estimates of the state of time t given past observations. They characterize the limit of their error sequence {ε_t, 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