Asymptotic spectrum of weighted sample covariance: another proof of spectrum convergence

TL;DR

This paper presents a new, concise proof of the convergence of the spectrum of weighted sample covariance matrices in high dimensions, including analysis for various weight distributions and heavy-tailed data.

Contribution

It offers a more streamlined proof with stronger assumptions and extends the understanding of spectrum behavior in finite samples with heavy tails.

Findings

Spectrum converges in high dimensions under weighted sampling

Spectrum behavior characterized for different weight distributions

Heavy-tailed data impacts spectrum in finite samples

Abstract

We propose another proof of the high dimensional spectrum convergence of the weighted sample covariance, more concise and self-sufficient but with stronger, but reasonable assumptions. We explain and illustrates this theorem for different weight distributions and show how the spectrum behaves in finite samples with heavy tails. The general purpose is to provide a detailed introduction to the high dimensional spectrum of weighted sample covariance.