CHAPTER V - LIMIT THEOREMS AND ESTIMATES OF THE MEAN

Abstract

This chapter discusses the central limit theorem and the role played by the normal law. In estimating relative frequencies and some other mean values, it is important that the distributions of these variables tend to the normal distribution. For the proof of these limit theorems, the Russian mathematician A.M. Lyapunov exploited the powerful method of characteristic functions that enabled him to prove the central limit theorem. A characteristic function determines a distribution function. If two random variables have identical characteristic functions, then they have the same distribution function. The characteristic function of a sum of independent random variables is equal to the product of the characteristic functions. The chapter discusses the connection between the characteristic function and the moments of a distribution. The characteristic function determines the probability distribution of the random variable; therefore, it follows that one can express all the moments in terms of the characteristic function.