Asymptotic analysis of non-linear system state probability distribution

In this paper, we study the problem of existence of limit distribution of the discrete-time stochastic process described by nonlinear difference stochastic equation. Our method differs from the standard ones based on stochastic modification of Lyapunov´s stability theory, and the ergodic theory. We consider the asymptotic behavior of the solution of stochastic differential equation as the convergence of related probability measure, similar to the classical theory of limit distributions. Our method is based on the contracting operators techniques in metric spaces. So, we can establish not only existence of the limit distribution, but, also the convergence of some distribution moments to their limit values, which is of valuable practical important