Linear spectral statistics of sequential sample covariance matrices

TL;DR

This paper establishes the weak convergence of linear spectral statistics of sequential sample covariance matrices in high dimensions, enabling new methods for testing sphericity when data dimension exceeds sample size.

Contribution

It introduces a novel asymptotic theory for sequential spectral statistics of high-dimensional covariance matrices, facilitating sphericity testing in large-dimensional settings.

Findings

Weak convergence of spectral process to Gaussian process

Applicable to high-dimensional sphericity testing

Method effective even when data dimension exceeds sample size

Abstract

Independent $p$-dimensional vectors with independent complex or real valued entries such that $\mathbb{E} [\mathbf{x}_i] = \mathbf{0}$, ${\rm Var } (\mathbf{x}_i) = \mathbf{I}_p$, $i=1, \ldots,n$, let $\mathbf{T }_n$ be a $p \times p$ Hermitian nonnegative definite matrix and $f $ be a given function. We prove that an approriately standardized version of the stochastic process $ \big ( {\operatorname{tr}} ( f(\mathbf{B}_{n,t}) ) \big )_{t \in [t_0, 1]} $ corresponding to a linear spectral statistic of the sequential empirical covariance estimator $$ \big ( \mathbf{B}_{n,t} )_{t\in [ t_0 , 1]} = \Big ( \frac{1}{n} \sum_{i=1}^{\lfloor n t \rfloor} \mathbf{T }^{1/2}_n \mathbf{x}_i \mathbf{x}_i ^\star \mathbf{T }^{1/2}_n \Big)_{t\in [ t_0 , 1]} $$ converges weakly to a non-standard Gaussian process for $n,p\to\infty$. As an application we use these results to develop a novel approach for monitoring the sphericity assumption in a high-dimensional framework, even if the dimension of the underlying data is larger than the sample size.