Convergence in Mean, in Distribution

Abstract

We begin this lesson by reviewing the convergences introduced in Lessons 7 and 8, Part II; then we define and illustrate two new types. All together we will have the following “modes” of convergence:

pointwise convergence, convergence almost surely,convergence a.s. completely, convergence in probability convergence in a mean, convergence in distribution.

Perhaps the last mode is the one most used in statistics. Additional properties, including interdependencies, will be discussed in the next two lessons. To simplify matters, we take all random variables to be real valued, finite a.s., defined on a common probability space [Ω,ℬ, P].