CLT for random quadratic forms based on sample means and sample covariance matrices

TL;DR

This paper establishes a central limit theorem for quadratic forms derived from sample means and covariance matrices in high-dimensional settings, using dimensional reduction techniques to handle large p and q.

Contribution

It introduces a novel application of dimensional reduction to derive CLT results for quadratic forms based on sample means and covariances in high dimensions.

Findings

CLT for quadratic forms under high-dimensional asymptotics

Dimensional reduction technique effectively simplifies the analysis

Results applicable when p/n approaches zero

Abstract

In this paper, we use the dimensional reduction technique to study the central limit theory (CLT) random quadratic forms based on sample means and sample covariance matrices. Specifically, we use a matrix denoted by $U_{p\times q}$, to map $q$-dimensional sample vectors to a $p$ dimensional subspace, where $q\geq p$ or $q\gg p$. Under the condition of $p/n\rightarrow 0$ as $(p,n)\rightarrow \infty$, we obtain the CLT of random quadratic forms for the sample means and sample covariance matrices.