Kolmogorov Three Series Theorem
# Statement
Let be a Probability Space
... ...
and let $(X_{n}){n=1}^{\infty}$ be a Sequence of independent Random Variables on . Then $\sum\limits{n=1}^{\infty} X_{n}$ converges Almost Surely If and Only IfProbability Space
there exist such thatIf and Only If
- $\sum\limits_{n=1}^{\infty} \text{Var}(X_{n} \mathbb{1}{|X{n}|\leq c}) < \infty$
- $\sum\limits_{n=1}^{\infty} \mathbb{E}(X_{n} \mathbb{1}{|X{n}|\leq c})$ converges.
# Proof
TODO ...
Link: Google Drive
Classes: ...
- See Thm 7.6.1 in Resnick - A Probability PathTODO
pg 226Resnick - A Probability Path