Central limit theorems

Abstract

The central limit theorem is a fundamental tool in statistics. In this section, various central limit theorems for mixing random variables are presented in order to point out the real differences with the independent case. This chapter is divided in three parts. In the first one we recall sufficient conditions for CLT to hold. In the second we recall results concerning convergence rates in CLT. In the last part we prove a CLT with a rate involving explicitly the dimension of the underlying space. This kind of result, first proved in Yurinskii (1977) for independent and identically distributed random variables, yields weak invariance principles. Dehling (1983), Doukhan, León & Portal (1987) and Doukhan & Portal (1987) prove similar results for dependent random variables.