Conditions for the local convergence of recursive stochastic procedures

We obtain the necessary and sufficient conditions for the almost sure convergence of recursive procedure (2.1) to the equilibrium stable state of the vector field $b(x)$. It is assumed that the trajectories of this procedure return a. s. into any neighbourhood of the equilibrium state. The convergence under this assumption is called local. Local convergence is studied for the cases of power (theorem 3.1) and subexponential (theorems 4.1 and 4.2) tails of distributions of random perturbations $\xi(t)$.