Kolmogorov's two-series theorem

Kolmogorov's Two-Series Theorem is a result from probability theory about the convergence of random series. It follows from Kolmogorov's inequality and is used in one proof of the Strong Law of Large Numbers.

Statement of the Theorem

Let (X_n)_n∈N be independent random variables with expected values E[X_n]=a_n and variances 𝕍ar(X_n)=σ_n², such that ∑^∞_n=1 a_n converges in ℝ and ∑^∞_n=1 σ_i² < ∞. Then ∑^∞_n=1 X_n converges in ℝ almost surely.