Kolmogorov Convergence Criterion

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 Centered Random Variables on $\Omega$. If $$\sum\limits{n=1}^{\infty} \mathbb{E} (X_{n}^{2}) < \infty,$$ then $$\sum\limits_{n=1}^{\infty} X_{n}$$ converges almost surely.

• Series
• Series Convergence
• Expectation
• Second Moment