= Strong law for uniformly L2-bounded uncorrelated random variables
{title2=$\frac1n\sum_{k=1}^n(X_k-\mathbb E X_k)\to0\text{ a.s.}$}
If $\sup_n\mathbb E|X_n|^2<\infty$ and the $X_n$ are pairwise <uncorrelated random variables>, then $n^{-1}\sum_{k=1}^n(X_k-\mathbb E X_k)\to0$ 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 $L^2$.
Back to article page
