Weak convergence of recursions

In this paper, we study the asymptotic distribution of a recursively defined stochastic process where are d-dimensional random vectors, b, Rd → Rd and σ: Rd → Rd × r are locally Lipshitz continuous functions, {εn} are r-dimensional martingale differences, and {an} is a sequence of constants tending to zero. Under some mild conditions, it is shown that, even when σ may take also singular values, {Xn} converges in distribution to the invariant measure of the stochastic differential equation where is a r-dimensional Brownian motion