Abstract
It is proved that under a specific condition (so-called condition $G_2 $) on the transition probability operator of a measurable stationary Markov process, a recursive kernel estimate of the initial density is convergent in quadratic mean.Assumptions on the differential stochastic equations driven by Brownian motion are derived under which the stationary solution satisfies condition $G_2 $.The above results are applied to solve a class of nonlinear identification problems.
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