CHAPTER 5 - Strong Limit Theorems for Independent Random Variables

Abstract

This chapter discusses the strong limit theorems for independent random variables. The strong limit theorems or the strong limit laws in probability refer to those theorems that deal with almost sure convergence of a sequence of random variables. The chapter presents some of the best known and most widely used theorems of this type. A valuable tool for the strong limit theorems is Kolmogorov's Inequalities. The chapter presents Kolmogorov's three series theorem that states that if {Xn } is a sequence of independent random variables, The series Χn converges almost surely if and only if for some constant c > 0. A major theorem in probability theory is a theorem that states that that if a series of independent random variables converges in law, then it converges almost surely. The major tools for proving this theorem are the Kolmogorov three series theorem explored in the chapter.