We consider the one-step prediction problem for discrete-time linear systems in correlated plant and observation noises, and non-gaussian initial conditions. We investigate the asymptotic behavior of the expected square {GREEK LETTER SUB t} of the difference between the MMSE and LMSE (or Kalman) estimates of the state given past observations. We characterize the limit of the error sequence ( {GREEK LETTER SUB t}, t = 0,1, ...} and obtain some related rates of convergence, with complete analysis being provided for the scalar case. The discussion is based on the explicit representations which were obtained by the authors in [,] for the MMSE and LMSE estimates, and which explicitly display the dependence of these quantities on the initial distribution.