The Relations between Several Kinds of Convergence of Random Variables

Abstract

In this paper, we study the convergence of random variables. We outline the definition and fundamental properties of various kinds convergence and study their relations. We focus on proving several theorem, these theorems show the relations between the different types of convergence, and we give some counter example to show that some relations are not true. Convergence in the r − thmean is the simplest in form, its relationship with others is the weakest. Convergence almost surely is the strongest, but it doesn’t imply convergence in the r −thmean. Both convergence almost surely and convergence in the r − th mean imply convergence in probability, convergence in probability implies convergence in distribution. Cauchy convergence can judge the convergence of sequences of random variables by using certain characteristics of sequences.