The Geometry of Spectral Fluctuations: On Near-Optimal Conditions for Universal Gaussian CLTs, with Statistical Applications

TL;DR

This paper develops a universal Gaussian CLT framework for spectral statistics of high-dimensional covariance matrices, accounting for nonclassical fluctuations and phase transitions, with applications to sphericity testing.

Contribution

It introduces the GHOST framework for Gaussian CLTs with structured fourth order effects and explicitly characterizes correction terms in spectral fluctuations.

Findings

Proves a Gaussian CLT with explicit mean and covariance corrections.

Develops a blockwise mixed radial model verifying the abstract assumptions.

Identifies a phase transition in fluctuation scale due to energy fluctuations.

Abstract

We study linear spectral statistics of high dimensional sample covariance matrices in a regime where the empirical spectral distribution remains governed by the classical sample covariance law but the fluctuation theory is nonclassical. Our starting point is a decomposition of the covariance of centered quadratic forms into a universal Gaussian part and a model dependent fourth order correction. This leads to an abstract framework, termed GHOST, for universal Gaussian central limit theorems under structured fourth order effects. Under this framework, we prove a Gaussian central limit theorem for linear spectral statistics, with explicit mean and covariance corrections determined by a bilinear fourth order kernel. Boundary examples show that the conditions are close to necessary for a broad universal Gaussian closure. We then develop a blockwise mixed radial model that verifies the abstract assumptions and makes the correction explicit. The correction splits into an entrywise fourth moment component and a lockwise energy fluctuation component. The latter may change the fluctuation scale, leading to a phase transition at the level of fluctuations. As an application, we study sphericity testing. Under the spherical null, the general correction collapses to a single scalar parameter, yielding a feasible data driven correction of John's test.