On empirical distribution function of high-dimensional Gaussian vector components with an application to multiple testing

Abstract

This paper presents a study of the asymptotical behavior of the empirical distribution function (e.d.f.) of Gaussian vector components, whose correlation matrix $\Gamma^{(m)}$ is dimension-dependent. By contrast with the existing literature, the vector is not assumed to be stationary. Rather, we make a “vanishing second order” assumption ensuring the covariance matrix $\Gamma^{(m)}$ is not too far from the identity matrix, while the behavior of the e.d.f. is affected by $\Gamma^{(m)}$ only through the sequence $\gamma_{m}=m^{-2}\sum_{i\neq j}\Gamma_{i,j}^{(m)}$, as $m$ grows to infinity. This result recovers some of the previous results for stationary long-range dependencies while it also applies to various, high-dimensional, non-stationary frameworks, for which the most correlated variables are not necessarily close to each other. Finally, we present an application of this work to the multiple testing problem, which was the initial statistical motivation for developing such a methodology.