Embedding the Finite Sampling Process at a Rate

Abstract

A huge body of if and only if theorems can be obtained based on certain strong embedding theorems for the partial sum process $\mathbb{S}_n$ and the uniform empirical and quantile processes $\mathbb{U}_n$ and $\mathbb{V}_n$. This embedding was accomplished in 1986 by M. Csorgo, S. Csorgo, L. Horvath and D. Mason. Their embedding is beautifully formulated so that many necessary and sufficient type results can be established using it. It is worthwhile to have an accessible proof. Indeed, these authors have since produced two papers that obtain the essential form of their result without appealing to earlier work of J. Komlos, P. Major and G. Tusnady. Indeed, this present paper does this for the finite sampling process $\mathbb{R}_n$ and the weighted empirical process $\mathbb{W}_n$. These latter results are entirely new and are the main objectives of the present paper. The applications of these latter results will appear elsewhere.