 Finite covariance range weak law

Suppose the variances of a sequence of random variables are bounded by $C$ and $\mathrm{Cov}(X_i,X_j)=0$ 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.