## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60G50: Sums of independent random variables; random walks

**Keywords**

Central limit theorem convergence determining sets independent and identically distributed variables rate of convergence

#### Citation

Hall, Peter. Sets Which Determine the Rate of Convergence in the Central Limit Theorem. Ann. Probab. 11 (1983), no. 2, 355--361. doi:10.1214/aop/1176993601. https://projecteuclid.org/euclid.aop/1176993601