Abstract
This chapter discusses the infinite sequences of random variables and their convergence. The various modes of stochastic convergence have a number of similarities with the convergence concept of classical analysis. The main difference is that all definitions of stochastic convergence take into account the existence of a probability measure. There are many relationships between the modes of stochastic convergence. The concepts of almost certain convergence and convergence in probability correspond to the measure-theoretic concepts of convergence almost everywhere and convergence in measure, respectively. These two modes of stochastic convergence are applicable to any sequence of random variables. Another convergence concept is the weak convergence of a sequence of distribution functions. The idea of weak convergence is sometimes carried over to the sequences of random variables. There are several modes of stochastic convergence that are different from each other, for example, almost certain convergence implies convergence in probability. However, the converse is not true.
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