Chapter 1 - ELEMENTS OF STOCHASTIC PROCESSES

Abstract

This chapter reviews the basic elementary notions and terminology of probability theory.The chapter discusses joint distribution functions, and reviews the marginal distribution function. The conditional distributions and densities, and infinite families of random variables are also reviewed. The relation between distribution functions and characteristic functions is one-to-one. The equation which expresses the distribution function in terms of its characteristic function is known as Levy's inversion formula. The one-to-one correspondence between distribution functions and their characteristic functions is also preserved by various limiting processes. Many of the basic results of probability theory are in the form of limit theorems. The Borel-Cantelli lemma is discussed in the chapter. The theory of stochastic processes is concerned with the investigation of the structure of families of random variables X t , where t is a parameter running over a suitable index set T . The values of the X t , may be one-dimensional, two-dimensional, or n -dimensional, or even more general.