Joint CLT for eigenvalue statistics from several dependent large dimensional sample covariance matrices with application

TL;DR

This paper establishes a joint central limit theorem for eigenvalue statistics of multiple dependent large-dimensional sample covariance matrices, enabling improved high-dimensional white noise testing in time series analysis.

Contribution

It introduces the first joint CLT for linear spectral statistics of several dependent sample covariance matrices, extending beyond the single-matrix case.

Findings

Derived a joint CLT for multiple covariance matrices eigenvalues.

Applied the CLT to high-dimensional white noise testing, showing faster computation.

Test maintains controlled size but has lower power compared to permutation tests.

Abstract

Let $\mathbf{X}_n=(x_{ij})$ be a $k \times n$ data matrix with complex-valued, independent and standardized entries satisfying a Lindeberg-type moment condition. We consider simultaneously $R$ sample covariance matrices $\mathbf{B}_{nr}=\frac1n \mathbf{Q}_r \mathbf{X}_n \mathbf{X}_n^*\mathbf{Q}_r^\top,~1\le r\le R$, where the $\mathbf{Q}_{r}$'s are nonrandom real matrices with common dimensions $p\times k~(k\geq p)$. Assuming that both the dimension $p$ and the sample size $n$ grow to infinity, the limiting distributions of the eigenvalues of the matrices $\{\mathbf{B}_{nr}\}$ are identified, and as the main result of the paper, we establish a joint central limit theorem for linear spectral statistics of the $R$ matrices $\{\mathbf{B}_{nr}\}$. Next, this new CLT is applied to the problem of testing a high dimensional white noise in time series modelling. In experiments the derived test has a controlled size and is significantly faster than the classical permutation test, though it does have lower power. This application highlights the necessity of such joint CLT in the presence of several dependent sample covariance matrices. In contrast, all the existing works on CLT for linear spectral statistics of large sample covariance matrices deal with a single sample covariance matrix ($R=1$).