Radius identification for infinite-second-moment targets

Establish whether the parent radius variables R_p converge weakly to a specified probability law ν with infinite second moment when the small-subspace energy condition holds and every deterministic rank-⌊αp⌋ projected sample covariance ESD converges to the radial Marchenko–Pastur law μ_{cα,ν}.

Background

The paper proves that a common proportional-rank projected spectral limit identifies the parent radius law under the small-subspace energy condition when the limiting spectral law has finite second moment. The proof uses a bootstrap that derives a finite second moment for every subsequential radius limit, but this argument does not extend to target laws with infinite second moment.

The unresolved issue is whether the same radius-identification conclusion remains valid for arbitrary heavy-tailed target laws. Once R_p converges to the specified ν, the paper’s moment-free one-rank characterization yields the radial quadratic-form condition, so the open issue is specifically the recovery of ν from projected spectra before radius convergence is known.