The Annals of Probability

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

Erich Haeusler

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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


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