The Central Limit Theorem---Its Applications in Pure and Applied Mathematics.

摘要

The Central Limit Theorem is one of the most remarkable results of the theory of probability. In its simplest form, the theorem states that the sum of a large number of independent observations from the same distribution has, under certain general conditions, an approximate normal distribution. Moreover, the approximation steadily improves as the number of observations increases. The theorem explains why many distributions tend to be close to normal distribution. The key ingredient is that the random variable being observed should be the sum or mean of many independent identically distributed random variables. This thesis seeks to help the reader understand the Central Limit Theorem (CLT) through applications. It further investigates how the CLT is used to approximate other distributions by normal distributions. Chapter 3 presents a new application of the Central Limit Theorem. It is used to show the large sample precision of the variance estimators, a result contained in the unpublished paper by W. R. Javier, Mathematics Dept., SU-BR, in 2003.