Statistica Sinica 4(1994), 361-372

ALMOST SURE CONVERGENCE OF STOCHASTIC

APPROXIMATION PROCEDURES

Gang Li

Ohmeda

Abstract: In this paper we investigate the convergence of the Robbins-Monro procedure Xn+1=Xnan(Y nα) . The following along with some related results are obtained.

Let ξj=YjM(Xj) be the error in the jth observation. A necessary and sufficient condition for the almost sure convergence of {Xn} is

a.s.

If {ξj} is an i.i.d. sequence, p≥1, j =0, and aj=j-1/p for j≥1, then the above is true if and only if E1|p <∞ .

Key words and phrases: Robbins-Monro procedure, almost sure convergence, martingale differences.