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A control variate is a random variable $C$ correlated with the estimator and with known mean $c_0$. For a fixed real coefficient $a$, define $Y_a=Y-a(C-c_0)$. Then $\mathbb{E}{Y}_a=\theta$, whileAssuming finite second moments and positive control variance, complete the square to findWhen $\mathrm{Var}(Y)>0$, the minimum is $\mathrm{Var}(Y)(1-{\rho }_{Y,C}^2)$. A constant control cannot reduce variance. A strongly correlated control, including a negatively correlated one with a negative coefficient, can substantially reduce it.

For multivariate control variates, write $Y_{\beta }=Y-{\beta }^T(C-\mathbb{E}C)$, $\Sigma =\mathrm{Cov}(C)$ and $c=\mathrm{Cov}(C,Y)$. Completing the multivariate square givesIf $\Sigma$ is singular, use its Moore-Penrose inverse. Indeed any null direction has a constant centered control combination and therefore zero covariance with $Y$, so $c$ lies in the range of $\Sigma$. The elementary unbiasedness statement assumes a deterministic coefficient; estimating it from the same simulation may introduce finite-sample bias, whereas an independently trained coefficient keeps the conditional unbiasedness argument valid.