Approximation Methods which Converge with Probability one

Let H (y x) be a family of distribution functions depending upon a real parameter x, and let M (x) = ^- y dH (y x) be the corresponding regression function. It is assumed M (x) is unknown to the experimenter, who is, however, allowed to take observations on H (y x) for any value x. Robbins and Monro 1 give a method for defining successively a sequence \xₙ\ such that xₙ converges to in probability, where is a root of the equation M (x) = and is a given number. Wolfowitz 2 generalizes these results, and Kiefer and Wolfowitz 3, solve a similar problem in the case when M (x) has a maximum at x =. Using a lemma due to Loeve 4, we show that in both cases xₙ converges to with probability one, under weaker conditions than those imposed in 2 and 3. Further we solve a similar problem in the case when M (x) is the median of H (y x).