On a Class of Stochastic Approximation Processes

Abstract

We are concerned with the asymptotic behavior of stochastic approximation processes of the Robbins-Monro type [9], the Kiefer-Wolfowitz type [7], and related types. Our main interest is establishing the asymptotic normality, under appropriate conditions, of processes of certain kinds. However, a number of results on convergence with probability one are also obtained as immediate consequences of a theorem needed in the work on asymptotic normality. Our results contain and extend some of the results in this area reported by Blum [1], Chung [3], Hodges and Lehmann [5], and others. In addition, we establish a number of results for cases not previously investigated. For instance, we show that the Kiefer-Wolfowitz processes are asymptotically normal under quite general conditions. The rapidity of convergence depends on the amount that the function $M$ (of Corollary 3.2) departs from symmetry in the neighborhood of the location of the maximum. We give results on convergence with probability one and asymptotic normality of stochastic approximation processes useful in connection with the problem of finding the location of the point of inflection of a function. For all cases in which we establish asymptotic normality we also show how the unknown quantities in the variance of the limiting normal distribution can be estimated. These results make possible the construction of asymptotic confidence intervals free of unknowns. Other results, too detailed to be summarized here, are established which also might be of interest in practical applications. Our method of procedure is as follows. We define a class of stochastic approximation processes, denoted by $A_0,$ which contains the class of Robbins-Monro processes, the class of Kiefer-Wolfowitz processes, and some other related classes. We first study the class $A_0$ using methods similar to those used in [1], [3], and [5]. After various results at this level have been obtained, the results for the special cases follow with an economy of effort.