WEAK-CONVERGENCE OF RECURSIONS

In this paper, we study the asymptotic distribution of a recursively d efined stochastic process Xn + 1 = X-n + a(n)(2)b(X-n) + a(n) sigma(X- n)epsilon(n + 1), where {X-n} are d-dimensional random vectors, b:R-d- ->R-d and sigma:R-d-->R-dxr are locally Lipshitz continuous functions, {epsilon(n)} are r-dimensional martingale differences, and {a(n)} is a sequence of constants tending to zero. Under some mild conditions, i t is shown that, even when a may take also singular values, (X,) conve rges in distribution to the invariant measure of the stochastic differ ential equation dZ(t) = b(Z(t))dt + sigma(Z(t)) dW(t), where W(t) is a r-dimensional Brownian motion.