Almost Sure Convergence under Estimating Conditional Mean Based on Dependent Data

The paper is devoted to establishing strong consistency of estimates of nonlinear characteristics of dynamic stochastic systems. To describe the shape of the nonlinearities, regression functions, i.e. conditional expectations of a variable with respect to another one, are used. In turn, the nonlinear regression functions are estimated by algorithms using the kernel-type approaches, which are suitable under fairly mild assumptions with respect to the system description. Within the approach, the key issue of the present paper is considering a case of mutually dependent observations in contrast to conventional nonparametric approaches based on regression estimates, which impose rather restrictive limitations on sampled data, e.g. mutual independence, various mixing conditions, etc., while such assumptions are not always acceptable within dynamic system considerations.