Radial Marchenko-Pastur laws: projection characterizations and rigidity

Abstract

Let \(R_p=\|x_p\|^2/p\Rightarrow \nu\) and \(p/N\to c\in(0,\infty)\). We prove that the radial quadratic-form condition (RQC), requiring energy in deterministic subspaces to follow the parent radius, is equivalent to the following spectral property: for one fixed \(\alpha\in(0,1)\), every deterministic rank-\(\lfloor \alpha p\rfloor\) projected covariance has empirical spectral distribution converging to \(\mu_{c\alpha,\nu}\). The radius law \(\nu\) may have arbitrary tails; no moment assumptions, conditional isotropy, or independence between radius and direction are required. We also show that the radius law can be recovered from the projected spectral limits when their common limit has finite second moment and subspaces of vanishing relative dimension carry vanishing normalized energy, again without moment assumptions on the original vectors. A finite-sample inverse estimate bounds quadratic-form defects by bounded tests of expected projected spectra. The converse argument combines angular symmetrization and fantope rigidity with a determinant comparison between radial column deletion and spectral trimming. A cancellation of the deleted radial factors makes the comparison error depend only on the deleted fraction, rather than on the magnitudes of the deleted radii.