Eigenvalue bounds in the critical regime

Prove the conjectured upper bound for the ordered eigenvalues of the asymptotic covariance matrix of a vector of normalized volume power functionals in the critical regime, assuming that the matrices \(A_m^{>1}\) satisfy Requirement 1, namely \(0\leq\lambda_1^{\Sigma_n}\leq\cdots\leq\lambda_n^{\Sigma_n}\leq (k_n+1)\max_{m=0,\ldots,k_n} S_m\left(\sum_{i=1}^n a_{i,m}^{-2}\right)\).

Background

In the critical regime, the covariance matrix is expressed as a finite weighted sum of matrices $A_m^{<1}$ or $A_m^{>1}$, depending on the value of the limiting parameter $c$. The paper derives upper bounds for the eigenvalues of each individual matrix $A_m^{>1}$ using spectral norms.

The proposed bound for the full covariance matrix requires Requirement 1, a Loewner-order domination condition that the paper notes may not generally be satisfied. The resulting statement is explicitly presented as a conjecture and is used later to obtain variance bounds for $h$-functionals.