Cauchy’s criterion on almost sure convergence
In this article, we will investigate interplays between various kinds of convergence of sequence of random variables. Specifically, we will discuss convergence a.s., convergence in probability, convergence in th mean and convergence in distribution.
We can naturally formalize the notions of sequence of random variables being Cauchy in these contexts of convergence. We say: is Cauchy in probability if as ; is Cauchy a.s. if is Cauchy everywhere except a set of measure zero; is Cauchy in if as . Now we give some basic facts about the advantages of using Cauchy’s criterion.
Proposition 1 a.s. if and only if for every as .
Proof: Let us consider the formulation of the set of points where ; that is
In fact, we can use the denseness of real numbers to write
Proposition 2 is Cauchy a.s. if and only if as for every ; or equivalently as for every .
Proof: Using a similar argument, we obtain a chain of equivalences,
where the last equivalence is obtain from the following inequality
This completes the proof.
Remark: Similarly one can formulate another equivalent definition of Cauchy a.s., that is if for all as , then is Cauchy a.s. Let . Then there is an integer such that whenever . Sending gives which is just the previous proposition.