## The Annals of Mathematical Statistics

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

#### Citation

Derman, Cyrus. An Application of Chung's Lemma to the Kiefer-Wolfowitz Stochastic Approximation Procedure. Ann. Math. Statist. 27 (1956), no. 2, 532--536. doi:10.1214/aoms/1177728277. https://projecteuclid.org/euclid.aoms/1177728277