## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 3 (1981), 395-404.

### Rates of Convergence in the Martingale Central Limit Theorem

#### Abstract

We obtain a nonuniform estimate of the rate of convergence in the martingale central limit theorem for convergence to mixtures of normal distributions. The uniform rates of convergence obtained by several other authors are special cases of our nonuniform estimate. We also obtain a rate of convergence in B. M. Brown's central limit theorem, assuming only Brown's elementary conditions. This result is a martingale analogue of Feller's generalization of the Berry-Esseen theorem.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 3 (1981), 395-404.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994413

**Digital Object Identifier**

doi:10.1214/aop/1176994413

**Mathematical Reviews number (MathSciNet)**

MR614625

**Zentralblatt MATH identifier**

0459.60042

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G42: Martingales with discrete parameter

**Keywords**

G0F05 Martingale central limit theorem mixtures of normal distributions nonuniform bound rate of convergence uniform bound

#### Citation

Hall, Peter; Heyde, C. C. Rates of Convergence in the Martingale Central Limit Theorem. Ann. Probab. 9 (1981), no. 3, 395--404. doi:10.1214/aop/1176994413. https://projecteuclid.org/euclid.aop/1176994413