Aug 2026· 1 citation· ⚡ 1 influential· 44 references
Mathematics
Abstract
We prove a central limit theorem for the log determinant of a Gaussian Pearson sample correlation matrix as the dimension diverges. Only two conditions are imposed: the population correlation matrix is positive definite, and the sample degrees of freedom are at least the dimension. Both are necessary for the ordinary log determinant to be finite. To the best of our knowledge, no previous central limit theorem covers this full nonsingular domain. It covers every aspect ratio from dilute growth to the square hard edge. No uniform lower or upper bound is imposed on the eigenvalues of the population correlation matrices: the smallest may approach zero and the largest may diverge. The proof develops a coordinatewise Wiener chaos reduction for the random diagonal normalization and combines it with an exact Wishart transform comparison. Geometrically, the statistic is twice the log volume of a random parallelotope spanned by standardized Gaussian coordinate vectors.
We study the annealed complexity of Gaussian random homogeneous polynomials on the (N−1)-dimensional unit sphere in the presence of deterministic perturbations depending on fixed orthonormal vectors and external parameters. We derive variational formulas for the exponential asymptotics of the average number of total cr...
Vanessa Piccolo· The Annals of Applied Probab...· 0 citations
We investigate the geometric fluctuations of principal subspaces for high-dimensional covariance matrices through the squared Frobenius $\sin\Theta$ distance between the sample and population eigenspaces associated with the $r_p$ largest eigenvalues. An explicit first-order expansion and a central limit theorem are est...
We study the asymptotic null spectrum of Gaussian-weighted tests of univariate normality with estimated location and scale. Closed Hermite coefficients give equations both away from and at the unperturbed poles. Every positive eigenvalue of the fully standardized covariance is simple, and the odd and even eigenvalues a...
N. Gkoumas, N. Papadatos, S. Trevezas· 0 citations
We study the microscopic spectrum at the origin of ratios of independent complex Girko matrices. Under bounded-density and finite-moment assumptions, together with circular second-moment matching, we prove compact-uniform convergence of the normalized determinants to a random entire function constructed from the comple...
We study moderate deviations for the largest eigenvalue of the randomly deformed Gaussian unitary ensemble introduced by Johansson (Probab. Theory Relat. Fields, {\bf 138}: 75--112, 2007). In the fixed-coupling regime, the rescaled largest eigenvalue converges to the convolution of the Tracy--Widom law and a Gaussian l...
It is well known in random matrix literature that the limiting spectral distribution of a Wigner matrix is the semi circular law while the limiting spectral distributions of other patterned matrices like Toeplitz, Hankel, symmetric circulant and reverse circulant matrices have unbounded supports. One fundamental differ...
