Central Limit Theorems

Introduction

One of the objectives of statistical inference is to draw conclusions about some parameter, like the mean or the variance of a (possibly conceptual) population of interest based on the information obtained in a sample conveniently selected therefrom. For practical purposes, estimates of these parameters must be coupled with statistical properties and except in the most simple cases, exact properties are difficult to obtain and one must rely on approximations. It is quite natural to expect estimators to be consistent, but it is even more important that their (usually mathematically complex) exact sampling distribution be adequately approximated by a simpler one, such as the normal or the χ ² distribution, for which tables or computational algorithms are available. Here we are not concerned with the convergence of the actual sequence of statistics {T _n } to some constant or random variable T as n → ∞, but with the convergence of the corresponding distribution functions {G...