Natural statistics for spectral samples

TL;DR

This paper develops spectral statistics analogous to classical sampling tools, providing unbiased estimators for matrix trace cumulants and linking spectral polykays to free cumulants in infinite populations.

Contribution

It introduces spectral k-statistics and normalized spectral polykays, extending classical statistical concepts to spectral sampling and free probability contexts.

Findings

Spectral k-statistics are unbiased estimators of trace cumulants.

Normalized spectral polykays relate to free cumulants in infinite populations.

The framework parallels classical sampling with spectral sampling in matrix groups.

Abstract

Spectral sampling is associated with the group of unitary transformations acting on matrices in much the same way that simple random sampling is associated with the symmetric group acting on vectors. This parallel extends to symmetric functions, k-statistics and polykays. We construct spectral k-statistics as unbiased estimators of cumulants of trace powers of a suitable random matrix. Moreover we define normalized spectral polykays in such a way that when the sampling is from an infinite population they return products of free cumulants.