Central limit theorems

Abstract

At the end of the 17th century, the mathematician Abraham de Moivre first used the normal distribution as an approximation for the percentage of successes in a large number of experiments. Later on, Laplace generalized his results, but it took 20th century mathematics to give an exact and complete description of this subject. So let me now describe the modern approach. We assume that for each n we have given a sequence X _{1, n},…X _n,n of random variables, which we assume to be independent. Then we want to ‘approximate’ the distribution of by a standard normal distribution, whose density equals.