Empirical Process of Multivariate Gaussian under General Dependence

Abstract: This paper explores certain kinds of empirical process with respect to the components of multivariate Gaussian. We put forward some finite sample bounds which hold for multivariate Gaussian under general dependence. We give necessary and sufficient condition for the convergence in probability of the random variable sequence $\displaystyle\left{\sup_t\vert\widehat{F}n(t)-\mathbf{E}\widehat{F}_n(t)\vert\right}{n\in \mathbb{N}}$, where $\widehat{F}_n(t)$ is the empirical distribution. Also, we find a similar sufficient condition for almost surely convergence.