Weak convergence of the empirical spectral distribution of ultra-high-dimensional banded sample covariance matrices

TL;DR

This paper studies the spectral distribution of high-dimensional banded covariance matrices as their size grows, showing it converges to a deterministic measure under specific asymptotic conditions.

Contribution

It establishes the weak convergence of the empirical spectral distribution for ultra-high-dimensional banded covariance matrices in a new asymptotic regime.

Findings

Empirical spectral distribution converges weakly to a deterministic measure.

The limiting measure is characterized by its moments.

Restricted compositions of natural numbers are key in moment evaluation.

Abstract

In this article we investigate high-dimensional banded sample covariance matrices under the regime that the sample size $n$, the dimension $p$ and the bandwidth $d$ tend simultaneously to infinity such that $$n/p\to 0 \ \ \text{and} \ \ 2d/n\to y>0.$$ It is shown that the empirical spectral distribution of those matrices almost surely converges weakly to some deterministic probability measure which is characterized by its moments. Certain restricted compositions of natural numbers play a crucial role in the evaluation of the expected moments of the empirical spectral distribution.