XII - Convergence of Series of Independent Random Variables

Abstract

This chapter discusses the convergence of series of independent random variables. The chapter presents the following four modes of convergence in probability theory: (1) almost sure convergence, (2) convergence in probability, (3) convergence in distribution, and (4) convergence in the mean Lp or convergence in the norm Lp, p > 0. In the terminology of real analysis, almost sure convergence is called almost everywhere convergence, and convergence in probability is called convergence in measure. The chapter also discusses the Borel theorem, Borel–Cantelli lemma, and the zero-one law. The three notions of convergence are identical for the convergence of a series of independent random variables. Once this result is achieved, the problem of the almost sure convergence of a series of independent random variables can be handled by the use of characteristic functions of random variables, as the convergence in distribution is completely in the category of the convergence of characteristic functions based on the Levy continuity theorem.