This paper introduces a new framework to study the asymptotical behavior of the empirical distribution function of Gaussian vector components, whose correlation matrix $\Gamma^{(m)}$ is dimension-dependent. Hence, by contrast with the existing literature, the vector is not assumed to be stationary. Rather, the covariance matrix $\Gamma^{(m)}$ should be close enough to the identity matrix as $m$ grows to infinity. Markedly, under this assumption, the convergence result depends on $\Gamma^{(m)}$ only through the sequence $\gamma_m=m^{-2} \sum_{i\neq j} \Gamma_{i,j}^{(m)}$. This result recovers some of the previous results for stationary long-range dependencies while it also applies to various non-stationary cases, for which the most correlated variables are not necessarily next 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.