= Gaussian rotation of independent copies
{c}
{title2=$(U_\theta,V_\theta)=(X\sin\theta+Y\cos\theta,X\cos\theta-Y\sin\theta)$}
If $X,Y$ are <independent and identically distributed random variables> with a centered <multivariate normal distribution> and <covariance matrix> $\Sigma$, then $(U_\theta,V_\theta)$ has the same <probability law> as $(X,Y)$. Its two marginal <covariance matrices> are $\Sigma$ and its cross-<covariance> is zero. The pair has a <multivariate normal distribution>, so the two components are <independent>. This remains valid when $\Sigma$ is singular.
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