When do nonlinear filters achieve maximal accuracy?

The nonlinear filter for an ergodic signal observed in white noise is said to achieve maximal accuracy if the stationary filtering error vanishes as the signal to noise ratio diverges. We give a general characterization of the maximal accuracy property in terms of various systems theoretic notions. When the signal state space is a finite set explicit necessary and sufficient conditions are obtained, while the linear Gaussian case reduces to a classic result of Kwakernaak and Sivan (1972). 1. Introduction. Let (X t) t≥0 be a signal of interest, which we model as an ergodic Markov process. It is often the case that the detection of such a signal is imperfect: only some function of the signal may be directly observable, and the observations are additionally corrupted by additive white noise. That is, one observes in practice the integrated observation process