On a Stochastic Approximation Method

Abstract

Asymptotic properties are established for the Robbins-Monro [1] procedure of stochastically solving the equation $M(x) = \alpha$. Two disjoint cases are treated in detail. The first may be called the "bounded" case, in which the assumptions we make are similar to those in the second case of Robbins and Monro. The second may be called the "quasi-linear" case which restricts $M(x)$ to lie between two straight lines with finite and nonvanishing slopes but postulates only the boundedness of the moments of $Y(x) - M(x)$ (see Sec. 2 for notations). In both cases it is shown how to choose the sequence $\{a_n\}$ in order to establish the correct order of magnitude of the moments of $x_n - \theta$. Asymptotic normality of $a^{1/2}_n(x_n - \theta)$ is proved in both cases under a further assumption. The case of a linear $M(x)$ is discussed to point up other possibilities. The statistical significance of our results is sketched.