Some modes of convergence of random variables, applications
The first thing which is done in this chapter is to introduce two modes of convergence for sequences of r.v.'s, convergence in distribution and convergence in probability, and then to investigate their relationship.
A suitable application of these convergences leads to the most important results in this chapter, which are the weak law of large numbers (WLLN) and the central limit theorem (CLT). These results are illustrated by concrete examples, including numerical examples in the case of the CLT.
In the final section of the chapter, it is shown that convergence in probability is preserved under continuity. This is also the case for convergence in distribution, but there ...
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