CHAPTER 3 - Stochastic Independence

Abstract

One of the most frequently used properties in probability theory is that of independence. This chapter discusses the stochastic independence and its properties. The assumption of independence is strong and yields strong results. One of the strongest results obtained from independence is a theorem by A. N. Kolmogorov, referred to as the zero-one law. The chapter presents a proof for the zero-one law and Borel–Cantelli lemma. The chapter describes and characterizes independence of random variables and deduces special properties. The random variables are independent if and only if those of every finite subset are independent. The random variables are independent if and only if the classes of events are independent. Characteristic functions become useful when the distribution function of a sum of independent random variables is to be found. A useful tool for studying sums of independent random variables is the concentration function. The concentration function shows the extent of concentration of the probability measure on the line. If two independent random variables are added, the spread of probability mass of their distribution extends over a wider range.