Maximum of sparsely equicorrelated Gaussian fields and applications

TL;DR

This paper studies the extreme values of a specific sparse, equicorrelated Gaussian field on a triangle, identifying the correlation threshold where the Gumbel law no longer applies, with implications for high-dimensional statistics.

Contribution

It determines the correlation threshold for the breakdown of the Gumbel law in a sparse Gaussian field and applies the findings to multiple testing and open problems in the field.

Findings

Identified the critical correlation parameter r where Gumbel law fails.

Applied Chen-Stein method for Poisson approximation in the analysis.

Provided insights into multiple testing and resolved open questions.

Abstract

We investigate the extreme values of a sparse and equicorrelated Gaussian field on a triangle: the correlations on every vertical or horizontal line are all equal to a parameter $r \in [0,1/2]$ and are zero everywhere else. This problem is closely linked with various problems in high-dimensional statistics and extreme-value theory. We identify the threshold for $r$ at which the standard Gumbel law breaks down. Our result is based on a subtle application of the Chen-Stein method for Poisson approximation. As applications, we discuss the implication of our results on multiple testing and resolve several questions that were left open in \cite{heiny2024maximum}, \cite{tang2022asymptotic} and \cite{Jiang19}.