Statistica Sinica

Gang Li

Abstract:In this paper we investigate the convergence of the Robbins-Monro procedureX=_{n+1}X—_{n}a(_{n}Y—_{ n}α) . The following along with some related results are obtained.

Letξ=_{j}Y—_{j}M(X) be the error in the_{j}jth observation. A necessary and sufficient condition for the almost sure convergence of {X} is_{n}

If {ξ_{j}} is an i.i.d. sequence,p≥1,Eε=0, and_{j}a=_{j}j^{-1/p}forj≥1, then the above is true if and only ifE|ξ_{1}|^{p}<∞ .

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