Non-Gaussianity after spectral compression: Fisher geometry and testing limits for Wishart matrices

This record contains the preprint “Non-Gaussianity after Spectral Compression: Fisher Geometry and Testing Limits for Wishart Matrices” by Surya Tallavarjula and Seyed Ali Rastegar. The paper studies how much information about the distribution of the entries of a random data matrix remains after the matrix is compressed to the eigenvalues of its Wishart sample covariance matrix. At the Gaussian Wishart reference model, we derive the local Fisher geometry of perturbations to the i.i.d. entry distribution and identify the exact spectral score associated with fourth-cumulant, or kurtosis, perturbations. The main results include: an exact finite-dimensional formula for the spectral Fisher information associated with the fourth cumulant; an explicit optimal score depending only on the first two spectral traces, (Tr(Y)) and (Tr(Y^2)); a hierarchy describing how much information about higher standardized cumulants survives spectral compression; an asymptotic rank-one collapse of the standardized tangent geometry onto the fourth-Hermite direction; a dimension-uniform (L^2) likelihood expansion for an explicit positive fourth-cumulant path; total-variation bounds and a one-spectrum ceiling for detecting small non-Gaussian perturbations; replicated local asymptotic normality without a coupling requirement between matrix dimension and the number of replications; local asymptotic sufficiency of the aggregated two-trace score; a Chebyshev decomposition of the limiting Fisher information into two independent spectral channels; exact first-order orthogonality between fourth-cumulant inference and covariance-scale nuisance directions; extensions to singular skewness experiments, sixth-cumulant inference, and Gaussian Orthogonal Ensemble models. The results connect random matrix theory, invariant statistical experiments, Fisher information, Wiener chaos, Haar integration, local asymptotic normality, and optimal hypothesis testing. They provide a quantitative account of which aspects of entrywise non-Gaussianity remain statistically recoverable from eigenvalues alone. The manuscript contains the complete mathematical arguments. Computational checks were used only for independent verification and do not support any load-bearing theorem or proof. Relevant verification code may be made available to researchers upon reasonable request. Authors:Surya Tallavarjula, Department of Physics, University of California, BerkeleySeyed Ali Rastegar, Department of Mathematics, University of California, Berkeley Funding: The authors received no external funding for this research. Status: Preprint submitted for peer review.