= Finite covariance range weak law
{title2=$\operatorname{Var}\!\left(\frac1n\sum_{i=1}^nX_i\right)\leq\frac{(2r+1)C}{n}$}
Suppose the <variances> of a sequence of <random variables> are bounded by $C$ and $\operatorname{Cov}(X_i,X_j)=0$ whenever $|i-j|>r$, for a fixed nonnegative integer $r$. The <Cauchy-Schwarz inequality> bounds each remaining covariance in absolute value by $C$. In the <variance of a sum>, there are at most $rn$ off-diagonal pairs, proving the displayed bound. The <Chebyshev inequality> makes the average minus its mean converge to zero in probability. If all means equal $\mu$, the average therefore converges in probability to $\mu$. Independence of every summand is unnecessary.
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