On the asymptotic distribution of the maximum sample spectral coherence of Gaussian time series in the high dimensional regime

TL;DR

This paper derives the asymptotic distribution of the maximum spectral coherence in high-dimensional Gaussian time series, enabling improved independence testing as both dimension and sample size grow large.

Contribution

It provides a new extreme value distribution result for spectral coherence in high dimensions, facilitating more accurate independence tests in large Gaussian time series.

Findings

Asymptotic distribution derived under specific growth conditions.

Supports independence testing with controlled significance levels.

Numerical simulations validate theoretical results.

Abstract

We investigate the asymptotic distribution of the maximum of a frequency smoothed estimate of the spectral coherence of a M-variate complex Gaussian time series with mutually independent components when the dimension M and the number of samples N both converge to infinity. If B denotes the smoothing span of the underlying smoothed periodogram estimator, a type I extreme value limiting distribution is obtained under the rate assumptions M N $\rightarrow$ 0 and M B $\rightarrow$ c $\in$ (0, +$\infty$). This result is then exploited to build a statistic with controlled asymptotic level for testing independence between the M components of the observed time series. Numerical simulations support our results.