Let $M(x)$ be a strictly increasing regression function for $x < \theta$, and strictly decreasing regression function for $x > \theta$. Under conditions 1, 2, and 3 given below, the stochastic approximation procedure proposed by Kiefer and Wolfowitz  is shown to converge stochastically to $\theta$. Under the additional conditions 4, 5, 6 given below, the procedure is shown to converge in distribution to the normal distribution. Our method is the one used by Chung .
Cyrus Derman. "An Application of Chung's Lemma to the Kiefer-Wolfowitz Stochastic Approximation Procedure." Ann. Math. Statist. 27 (2) 532 - 536, June, 1956. https://doi.org/10.1214/aoms/1177728277