Adaptive truncated covariance has a finite matrix-variance operator bound

Assume Countable Choice. Let $X_1,\dots ,X_m\in {\mathbb{R}}^p$ be clean iid observations with mean $\mu$, nonzero covariance $\mathrm{\Sigma }$, and finite fourth central moment. For the matrix variance in Matrix variance for covariance estimation, put $d=\frac{{\sigma }_0^2}{\mathrm{\parallel }\mathrm{\Sigma }{\mathrm{\parallel }}^2},D=\max \{d,{\left(\frac{\mathrm{tr}\mathrm{\Sigma }}{{\sigma }_0}\right)}^2\}\mathrm{.}$ Suppose $\beta >1$, $0<{\sigma }_{\min }\le {\sigma }_0\le {\sigma }_{\max }$ and $m\ge {10}^{16}D\beta \mathrm{.}$ The same-sample pilot and original scale/selection rule of Adaptive truncated covariance estimator then satisfy $P\text{\hspace{0.17em}⁣}(\mathrm{\parallel }{\widehat{\mathrm{\Sigma }}}^{\ast }-\mathrm{\Sigma }{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}>18{\sigma }_0\sqrt{\frac{\beta }{m}})\le 5p{\log }_2\text{\hspace{0.17em}⁣}\left(\frac{2{\sigma }_{\max }}{{\sigma }_{\min }}\right)e^{-\beta }\mathrm{.}$ Here $p$ is ambient dimension; it is not the intrinsic ratio $d$. More precisely, with $L_g={\log }_2(2{\sigma }_{\max }\mathrm{/}{\sigma }_{\min })$ and $N_{+}=\lceil L_g\rceil -\lceil {\log }_2({\sigma }_0\mathrm{/}{\sigma }_{\min })\rceil$, the failure bound is $[1+\frac{3p+1}{2}{N}_{+}]e^{-\beta }\mathrm{.}$ The trace-aware sample condition is sufficient, not necessary. All clean finite-fourth-moment laws remain eligible; no coordinate independence, kurtosis bound, contamination or replacement guarantee is imposed. The broader uniform pilot-accuracy formula in Fixed-scale matrix-truncated covariance deviation under finite matrix variance can be used instead of this sufficient sample condition whenever it establishes the same fixed-scale radius for each required scale.

The former $m\ge Cd\beta$ assertion for this unknown-mean estimator is false under arbitrary dependence, as proved by Intrinsic matrix variance alone does not control pilot-centered covariance truncation ↗. The separate two-axis example refutes the trace comparison used by the source. A full known-mean finite-matrix-variance concentration bound, without the trace-aware sample condition, is retained in Oracle-truncated covariance and vector concentration under finite matrix variance.