## The Annals of Probability

- Ann. Probab.
- Volume 16, Number 1 (1988), 275-299.

### On the Rate of Convergence in the Central Limit Theorem for Martingales with Discrete and Continuous Time

#### Abstract

Heyde and Brown (1970) established a bound on the rate of convergence in the central limit theorem for discrete time martingales having finite moments of order $2 + 2\delta$ with $0 < \delta \leq 1$. In the present paper a modification of the methods developed by Bolthausen (1982) is applied to show the validity of this result for all $\delta > 0$. Moreover, an example is constructed demonstrating that this bound is asymptotically exact for all $\delta > 0$. The result for discrete time martingales is then used to derive the corresponding bound on the rate of convergence in the central limit theorem for locally square integrable martingales with continuous time.

#### Article information

**Source**

Ann. Probab., Volume 16, Number 1 (1988), 275-299.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991901

**Digital Object Identifier**

doi:10.1214/aop/1176991901

**Mathematical Reviews number (MathSciNet)**

MR920271

**Zentralblatt MATH identifier**

0639.60030

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60G42: Martingales with discrete parameter 60G44: Martingales with continuous parameter

**Keywords**

Martingales with discrete and continuous time central limit theorem rate of convergence

#### Citation

Haeusler, Erich. On the Rate of Convergence in the Central Limit Theorem for Martingales with Discrete and Continuous Time. Ann. Probab. 16 (1988), no. 1, 275--299. doi:10.1214/aop/1176991901. https://projecteuclid.org/euclid.aop/1176991901