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May, 1983 Sets Which Determine the Rate of Convergence in the Central Limit Theorem
Peter Hall
Ann. Probab. 11(2): 355-361 (May, 1983). DOI: 10.1214/aop/1176993601


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.


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Peter Hall. "Sets Which Determine the Rate of Convergence in the Central Limit Theorem." Ann. Probab. 11 (2) 355 - 361, May, 1983.


Published: May, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0513.60032
MathSciNet: MR690133
Digital Object Identifier: 10.1214/aop/1176993601

Primary: 60F05
Secondary: 60G50

Keywords: central limit theorem , convergence determining sets , independent and identically distributed variables , rate of convergence

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 2 • May, 1983
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