Almost Sure Convergence for the Robbins-Monro Process

Abstract

In this paper we investigate the almost sure convergence of the Robbins-Monro process $x_{n+1} = x_n - a_n(y_n - \alpha)$ under assumptions about the conditional distribution of $y_n$ given $x_n$ which involve the existence of first moments or something closely related. The process $x_n$ can converge almost surely even when the series $\sum^\infty_{n=1} a_n\lbrack y_n - E\{y_n\mid x_n\} \rbrack$ does not do so.