Asymptotics of the Inertia Moments and the Variance Conjecture in Schatten Balls

TL;DR

This paper investigates the asymptotic behavior of moments of the Hilbert-Schmidt norm in p-Schatten balls, establishing a generalized variance conjecture for certain matrix families as dimension grows.

Contribution

It provides the first and second order asymptotic expansions of these moments and extends the variance conjecture to p-Schatten unit balls of self-adjoint matrices for p > 3.

Findings

Asymptotic expansions of moments as dimension increases

Validation of a generalized variance conjecture for p > 3

Applicability to real, complex, and quaternionic matrices

Abstract

We study the first and second orders of the asymptotic expansion, as the dimension goes to infinity, of the moments of the Hilbert-Schmidt norm of a uniformly distributed matrix in the p-Schatten unit ball. We consider the case of matrices with real, complex or quaternionic entries, self-adjoint or not. When p > 3, this asymptotic expansion allows us to establish a generalized version of the variance conjecture for the family of p-Schatten unit balls of self-adjoint matrices.