ON DYNAMIC KIEFER-WOLFOWITZ STOCHASTIC APPROXIMATION PROCEDURES

Let M(x) be a function from Rk--4R.Let en, n = 1,2,... be vectors in Rk, el being the value at which M(x) achieves its
unique minimum. Set M(x) = Mi(x), for n = 1,2,..., set Mn(x) = M(x - en - el). Then en is the unique minimum of Mn(x), which is unknown and is to be estimated. In our case, we assume that en moves in such a manner that en+1 = gn(8n) + vn where gnn) is general non-linear k-vector measurable function (known) defined for all >cellk and vn is an unknown k-vector function (random or non-random) independent of x. Let an, cn, n = 1,2,... be two sequences of positive numbers. Let xl be an arbitrary random variable. Define for n = 1,2,...,xn+1 = xn an(Y2n - y2n-1)/cn where xn = gn(xn), and v2n' Y2n-1 are random variables such that - their expectations given xi,x2, ,xn are E Mn+1(xn + eicn)e i-1 k and E Mn+1(xn e c n)e. respectively and their conditional 1 1=1 variance are bounded by a constant a2 and they are conditionally
independent. Under conditions similar to those used by Dupac (1966), we show that Ilxn - en 0 with probability one.