 Gaussian sample-covariance operator-norm bound

Let $X_1,\dots ,X_n\in {\mathbb{R}}^p$ be independent centered Gaussian random vectors with common nonzero covariance matrix $\Sigma$, and put $\hat{\Sigma }=n^{-1}{\sum }_i{X}_i{X}_i^T$. There is a universal constant $C>0$ such that for $t\ge 1$, with probability at least $1-e^{-t}$,where $r(\Sigma )=\mathrm{tr}(\Sigma )/\Vert \Sigma {\Vert }_{\mathrm{op}}$ 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.^[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 $\Sigma$.