Kolmogorov Three Series 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 Almost Surely If and Only If there exist $c > 0$ such that

1. $\sum\limits_{n=1}^{\infty} \mathbb{P}(|X_{n}| > c) < \infty$
2. $\sum\limits_{n=1}^{\infty} \text{Var}(X_{n} \mathbb{1}{|X{n}|\leq c}) < \infty$
3. $\sum\limits_{n=1}^{\infty} \mathbb{E}(X_{n} \mathbb{1}{|X{n}|\leq c})$ converges.

TODO - See Thm 7.6.1 in Resnick - A Probability Path pg 226

• Real Numbers
• Variance
• Expectation
• Indicator Function
• Series