= Gaussian sample-covariance operator-norm bound
Let $X_1,\ldots,X_n\in\mathbb R^p$ be <independent> centered <Gaussian random vectors> with common nonzero <covariance matrix> $\Sigma$, and put $\widehat\Sigma=n^{-1}\sum_iX_iX_i^T$. There is a universal constant $C>0$ such that for $t\geq1$, with <probability> at least $1-e^{-t}$,
$$
\|\widehat\Sigma-\Sigma\|_{\mathrm{op}}\leq C\|\Sigma\|_{\mathrm{op}}\left(\sqrt{\frac{r(\Sigma)+t}{n}}+\frac{r(\Sigma)+t}{n}\right),
$$
where $r(\Sigma)=\operatorname{tr}(\Sigma)/\|\Sigma\|_{\mathrm{op}}$ is the <effective rank of a covariance matrix>.
The bound combines an expected <operator norm> error with <Gaussian concentration inequality>. In Euclidean space, the Gaussian effective-dimension parameter in the primary result is comparable to this trace ratio; replacing maxima by sums gives the displayed version.<Concentration Inequalities and Moment Bounds for Sample Covariance Operators by Vladimir Koltchinskii and Karim Lounici>[Definition 1 and discussion of trace effective rank; Corollary 2, equation 2.8]{ref}
For $\Sigma=I_p$, the <covariance effective rank> is $p$, and relative error is controlled by $\sqrt{(p+t)/n}+(p+t)/n$. <covariance> concentrated in fewer directions can have much smaller <covariance effective rank>. The estimator here is the uncentered empirical second moment for a known zero <mean>; an unknown <mean> and empirical centering require the corresponding formulation. The bound remains meaningful when $\Sigma$ is singular but nonzero, since it does not invert $\Sigma$.
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