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Matrix variance need not dominate trace covariance times operator norm

Statement refuted

For every centered square-integrable random vector $Z$ with finite fourth moment, the matrix variance $σ_{0}^{2} = {∥ E [∥ Z ∥_{2}^{2} Z Z^{T}] ∥}_{o p}$ is at least $tr (Σ) ∥ Σ ∥_{o p}$ , where $Σ = E [Z Z^{T}]$ .

Facts & Assumptions

Given: The covariance, matrix-variance, and operator-norm definitions in Covariance matrix of a finite-dimensional random vector, Matrix variance for covariance estimation, and The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||.

[F1]

Covariance matrix of a finite-dimensional random vector: for a centered vector, $Σ = E [Z Z^{T}]$ .

[F2]

Matrix variance for covariance estimation: $σ_{0}^{2} = ∥ E [∥ Z ∥_{2}^{2} Z Z^{T}] ∥_{o p}$ .

[F3]

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: the operator norm is the maximum of $∥ A v ∥_{2}$ over unit vectors.

[F4]

Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis: a real symmetric matrix has an orthonormal eigenbasis, so the operator norm of a diagonal symmetric matrix is the largest absolute diagonal entry.

Counterexample

1.1givenF1algebra

In $R^{2}$ , let $Z$ take values $\pm \sqrt{4 / 3} e_{1}$ with total probability $3 / 4$ (each sign has probability $3 / 8$ ), and $\pm \sqrt{2} e_{2}$ with total probability $1 / 4$ (each sign has probability $1 / 8$ ). Symmetry gives $E Z = 0$ , and finite support gives a finite fourth moment.

1.2givenF1F4algebra

Direct summation and [F1] give $Σ = E [Z Z^{T}] = diag (1, 1 / 2)$ , hence $tr (Σ) = 3 / 2$ and $∥ Σ ∥_{o p} = 1$ by [F4].

2.1F2F3F4step 1.2algebra∎

The value of $∥ Z ∥_{2}^{2}$ is $4 / 3$ on the first pair and $2$ on the second, so $E [∥ Z ∥_{2}^{2} Z Z^{T}] = diag ⁣ (\frac{3}{4} {(\frac{4}{3})}^{2}, \frac{1}{4} (2)^{2}) = diag (4 / 3, 1) .$ Its operator norm is $4 / 3 < 3 / 2 = tr (Σ) ∥ Σ ∥_{o p}$ by [F4]. Thus the statement is false.

Remarks

This refutes the auxiliary inequality used in Minsker–Wei, Corollary 5.1, for arbitrary dependent coordinates. It does not by itself refute their covariance-estimator theorem; another argument or corrected hypotheses would be needed.

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Sources

Stanislav Minsker and Xiaohan Wei, Estimation of the Covariance Structure of Heavy-Tailed Distributions, Corollary 5.1 (standard reference, not scraped)