Kolmogorov's two series theorem

Statement of the theorem

Let ( X n ) n = 1 ∞ be independent random variables with expected values E [ X n ] = μ n and variances V a r ( X n ) = σ n 2 , such that ∑ n = 1 ∞ μ n converges in ℝ and ∑ n = 1 ∞ σ n 2 converges in ℝ. Then ∑ n = 1 ∞ X n converges in ℝ almost surely.

Proof

Assume WLOG μ n = 0 . Set S N = ∑ n = 1 N X n , and we will see that lim sup N S N − lim inf N S N = 0 with probability 1.

For every m ∈ N ,

Thus, for every m ∈ N and ϵ > 0 ,

While the second inequality is due to Kolmogorov's inequality.

By the assumption that ∑ n = 1 ∞ σ n 2 converges, it follows that the last term tends to 0 when m → ∞ , for every arbitraty ϵ > 0 .