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 ₙ\ such that xₙ converges to θ in probability, where θ is a root of the equation M(x) = α and α 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 = θ. Using a lemma due to Loeve [4], we show that in both cases xₙ converges to θ 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).