A lower and upper bound on the a priori optimal mean square error is used to study the stationary behavior of one dimensional nonlinear filters. The long time behavior as t--> INFINITY for asymptotically linear systems is investigated. Lower and upper bounds of the Riccati type are derived and it is shown that for nonlinear systems with linear limiting ones, the Kalman filter (KF) formally designed for the limiting systems is asymptotically optimal in some sense. Examples with simulation results are provided.