## The Annals of Mathematical Statistics

### An Application of Chung's Lemma to the Kiefer-Wolfowitz Stochastic Approximation Procedure

Cyrus Derman

#### Abstract

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 [3] 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 [2].

#### Article information

Source
Ann. Math. Statist., Volume 27, Number 2 (1956), 532-536.

Dates
First available in Project Euclid: 28 April 2007

https://projecteuclid.org/euclid.aoms/1177728277

Digital Object Identifier
doi:10.1214/aoms/1177728277

Mathematical Reviews number (MathSciNet)
MR78595

Zentralblatt MATH identifier
0074.35502

JSTOR