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This paper introduces a new framework to study the asymptotical behavior ofthe empirical distribution function (e.d.f.) of Gaussian vector components,whose correlation matrix $\Gamma^{(m)}$ is dimension-dependent. Hence, bycontrast with the existing literature, the vector is not assumed to bestationary. Rather, we make a "vanishing second order" assumption ensuring thatthe 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 thesequence $\gamma_m=m^{-2} \sum_{i\neq j} \Gamma_{i,j}^{(m)}$, as $m$ grows toinfinity. This result recovers some of the previous results for stationarylong-range dependencies while it also applies to various, high-dimensional,non-stationary frameworks, for which the most correlated variables are notnecessarily next to each other. Finally, we present an application of this workto the multiple testing problem, which was the initial statistical motivationfor developing such a methodology.

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