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.