## The Annals of Probability

### Sets Which Determine the Rate of Convergence in the Central Limit Theorem

Peter Hall

#### Abstract

Rates of convergence in the central limit theorem are frequently described in terms of the uniform metric. However, statisticians often apply the central limit theorem only at symmetric pairs of isolated points, such as the 5% points of the standard normal distribution, $\pm 1.645$. In this paper we study rates of convergence on sets of the form $\{-\theta, \theta\}$, where $\theta \geq 0$. It is shown that the rate of convergence on the 5% points is the same as the rate uniformly on the whole real line, up to terms of order $n^{-1/2}$. Curiously, the rate of convergence on the 1% points $\pm 2.326$ can be faster than the rate on the whole real line.

#### Article information

Source
Ann. Probab., Volume 11, Number 2 (1983), 355-361.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993601

Digital Object Identifier
doi:10.1214/aop/1176993601

Mathematical Reviews number (MathSciNet)
MR690133

Zentralblatt MATH identifier
0513.60032

JSTOR