Strong law for uniformly L2-bounded uncorrelated random variables ($\frac{1}{n}{\sum }_{k=1}^n(X_k-\mathbb{E}{X}_k)\to 0\text{ a.s.}$)

If ${\sup }_n\mathbb{E}|X_n{|}^2<\infty$ and the $X_n$ are pairwise uncorrelated random variables, then $n^{-1}{\sum }_{k=1}^n(X_k-\mathbb{E}{X}_k)\to 0$ almost surely. Independence is unnecessary for this version of the strong law of large numbers. A uniformly bounded martingale difference sequence meets its hypotheses, since the differences have mean zero and are orthogonal in $L^2$.