Levy's Theorem

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space and let $({X}{n}){n=1}^{\infty}$ be a Sequence of independent Random Variables on $\Omega$. Then $$\sum\limits_{n=1}^{\infty} X_{n}$$ converges in probability If and Only If it converges almost surely.

TODO - See thm 7.3.2 in Resnick - A Probability Path pg 211