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).

💡 Research Summary

The paper addresses a fundamental question in stochastic filtering: under what circumstances does a nonlinear filter attain “maximal accuracy,” meaning that the stationary filtering error disappears as the signal‑to‑noise ratio (SNR) becomes infinite? The authors consider an ergodic Markov signal (X_t) observed through additive white noise, i.e.
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