An asymptotically optimal nonparametric procedure for nonlinear filtering of stationary sequences with unknown static responses

A nonparametric procedure is proposed for nonlinear filtering of Markov stationary, in a narrow sense, random sequences whose state equations and family of finite-dimensional distributions are unknown. This procedure develops from a single realization of the observed process with strong mixing and some assumptions on noise distribution in observations. Asymptotically, or as the realization length increases, the performance of such a procedure is different by arbitrary $\varepsilon>0$ from optimal nonlinear filtering which is obtained with completely available static characteristics of unobserved signals. Examples are considered and results of a model experiment are reported.