Empirical Process of Multivariate Gaussian under General Dependence

TL;DR

This paper investigates the empirical process of multivariate Gaussian distributions under general dependence, providing finite sample bounds and conditions for convergence in probability and almost sure convergence.

Contribution

It introduces new finite sample bounds and necessary and sufficient conditions for convergence of empirical processes of multivariate Gaussian variables with general dependence.

Findings

Finite sample bounds for Gaussian empirical processes

Necessary and sufficient conditions for convergence in probability

Sufficient conditions for almost sure convergence

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.