Nonparametric methods in linear filtering of stationary random sequences

A nonlinear filtering technique is proposed for application to a stationary, in the narrow sense, ergodic random Markov sequences which is observed against the background noise. The state equation and a family of finite dimensional distributions of the useful signal are assumed unknown. The model of noise observation and distributions are assumed such that the conditional observation distribution density with a fixed useful signal belongs to an exponential family. R.m.s. convergence in an argument is proved of nonparametric estimates of the multi-dimensional probability density and its gradient in a uniform metrics.