Largest Eigenvalues of Principal Minors of Deformed Gaussian Orthogonal Ensembles and Wishart Matrices

TL;DR

This paper investigates the asymptotic distribution of the maximum eigenvalues of principal minors of high-dimensional Wishart matrices, revealing a Gumbel distribution for certain moments and a new distribution beyond that, with implications for signal processing and statistics.

Contribution

It introduces the analysis of maximum eigenvalues of principal minors of Wishart matrices and connects this to a deformed GOE, providing new distributional results and a novel proof approach.

Findings

Maximum eigenvalues follow Gumbel distribution when η is between 0 and 3.

A new distribution emerges for η exceeding 3.

Methodology combines Stein-Poisson approximation, conditioning, and U-statistics.

Abstract

Consider a high-dimensional Wishart matrix $\bd{W}=\bd{X}^T\bd{X}$ where the entries of $\bd{X}$ are i.i.d. random variables with mean zero, variance one, and a finite fourth moment $\eta$. Motivated by problems in signal processing and high-dimensional statistics, we study the maximum of the largest eigenvalues of any two-by-two principal minors of $\bd{W}$. Under certain restrictions on the sample size and the population dimension of $\bd{W}$, we obtain the limiting distribution of the maximum, which follows the Gumbel distribution when $\eta$ is between 0 and 3, and a new distribution when $\eta$ exceeds 3. To derive this result, we first address a simpler problem on a new object named a deformed Gaussian orthogonal ensemble (GOE). The Wishart case is then resolved using results from the deformed GOE and a high-dimensional central limit theorem. Our proof strategy combines the Stein-Poisson approximation method, conditioning, U-statistics, and the H\'ajek projection. This method may also be applicable to other extreme-value problems. Some open questions are posed.