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 X_n+1=X_n—a_n(Y _n—α) . The following along with some related results are obtained.

Let ξ_j=Y_j—M(X_j) be the error in the jth observation. A necessary and sufficient condition for the almost sure convergence of {X_n} is

a.s.

If {ξ_j} is an i.i.d. sequence, p≥1, Eε_j =0, and a_j=j^-1/p for j≥1, then the above is true if and only if E|ξ₁|^p <∞ .

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