AN ALMOST SURE CONVERGENCE THEOREM IN A STOCHASTIC APPROXIMATION METHOD WITH DEPENDENT RANDOM VARIABLES

This paper is a continuation of our papers [9], [10] and [11] and is concerned with a Robbins-Monro type stochastic approximation method when a sequence of dependently distributed random vectors is given. The method of stochastic approximation has been first proposed by H. Robbins and S. Monro ([5]) and its modifications have been thereafter given by many authors. A typical one of them is as follows. Suppose that an RN-valued random vector Y„(x) can be observed at xERN and each instant n, and the expected value of Y n(X), denoted by E[Y„(x)]=1/17,(x), is unknown to us. Assuming that the equation itin(x)=0 has a solution x=8,, for each n=1, 2, • , it is desire to estimate 0,, for sufficiently large n on the basis of observed values Y1(X0), Y2(Xi), Y.+I(X,i), ••• at the points X„ Xi, •-• , X„, ••• which are produced by the following recurrence relation,