Coherence of high-dimensional random matrices in a Gaussian case : application of the Chen-Stein method

TL;DR

This paper analyzes the asymptotic distribution of the coherence in high-dimensional Gaussian matrices using the Chen-Stein method, showing convergence to a Gumbel distribution and extending previous results.

Contribution

It generalizes previous work on coherence by deriving the limiting law in a Gaussian setting with bandwise covariance matrices using the Chen-Stein method.

Findings

Limiting law of normalized coherence is Gumbel distribution.

Numerical simulations confirm asymptotic behavior in high dimensions.

Provides insights into high-dimensional correlation matrix properties.

Abstract

This paper studies the $\tau$-coherence of a (n x p)-observation matrix in a Gaussian framework. The $\tau$-coherence is defined as the largest magnitude outside a diagonal bandwith of size $\tau$ of the empirical correlation coefficients associated to our observations. Using the Chen-Stein method we derive the limiting law of the normalized coherence and show the convergence towards a Gumbel distribution. We generalize here the results of Cai and Jiang [CJ11a]. We assume that the covariance matrix of the model is bandwise. Moreover, we provide numerical considerations highlighting issues from the high dimension hypotheses. We numerically illustrate the asymptotic behaviour of the coherence with Monte-Carlo experiment using a HPC splitting strategy for high dimensional correlation matrices.