About limiting spectral distributions of block-rescaled empirical covariance matrices

TL;DR

This paper studies the spectral distribution of block-rescaled empirical covariance matrices, showing it converges to an arcsine law under specific conditions and proposing a conjecture for more general ratios.

Contribution

It establishes the limiting spectral distribution as an arcsine law for certain ratios and independence conditions, and introduces a conjecture for other ratios.

Findings

Spectral distribution converges to arcsine law when ratio approaches 1.

Independence of samples in each block is crucial for the result.

Proposes a conjecture for cases with constant ratio in (0,1).

Abstract

We establish that the limiting spectral distribution of a block-rescaled empirical covariance matrix is an arcsine law when the ratio between the dimension and the underlying sample size converges to 1 and when the samples corresponding to each block are independent. We further propose a conjecture for the cases where the latter ratio converges to a constant in the unit interval.