Correlation tests and sample spectral coherence matrix in the high-dimensional regime

TL;DR

This paper establishes central limit theorems for spectral coherence matrix statistics in high-dimensional Gaussian time series, enabling improved independence testing with controlled error rates.

Contribution

It introduces new CLTs for spectral coherence matrix statistics in high-dimensional regimes, facilitating hypothesis testing of component independence.

Findings

Two new CLTs for spectral coherence statistics are validated.

The proposed tests effectively control asymptotic levels.

Numerical simulations demonstrate test performance.

Abstract

It is established that the linear spectral statistics (LSS) of the smoothed periodogram estimate of the spectral coherence matrix of a complex Gaussian high-dimensional times series (yn) n$\in$Z with independent components satisfy at each frequency a central limit theorem in the asymptotic regime where the sample size N , the dimension M of the observation, and the smoothing span B both converge towards +$\infty$ in such a way that M = O(N $\alpha$ ) for $\alpha$ < 1 and M B $\rightarrow$ c, c $\in$ (0, 1). It is deduced that two recentered and renormalized versions of the LSS, one based on an average in the frequency domain and the other one based on a sum of squares also in the frequency domain, and both evaluated over a well-chosen frequency grid, also verify a central limit theorem. These two statistics are proposed to test with controlled asymptotic level the hypothesis that the components of y are independent. Numerical simulations assess the performance of the two tests.