Geodesic
Conditions for the local convergence of recursive stochastic procedures
Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 1, pp. 135-142

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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 $\xi(t)$. 
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     author = {V. V. Godovan\v{c}uk and A. P. Korostelev},
     title = {Conditions for the local convergence of recursive stochastic procedures},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {135--142},
     publisher = {mathdoc},
     volume = {28},
     number = {1},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1983_28_1_a8/}
}
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V. V. Godovančuk; A. P. Korostelev. Conditions for the local convergence of recursive stochastic procedures. Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 1, pp. 135-142. http://geodesic.mathdoc.fr/item/TVP_1983_28_1_a8/