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April, 1991 Embedding the Finite Sampling Process at a Rate
Galen R. Shorack
Ann. Probab. 19(2): 826-842 (April, 1991). DOI: 10.1214/aop/1176990453


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.


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Galen R. Shorack. "Embedding the Finite Sampling Process at a Rate." Ann. Probab. 19 (2) 826 - 842, April, 1991.


Published: April, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0731.60026
MathSciNet: MR1106288
Digital Object Identifier: 10.1214/aop/1176990453

Primary: 60F17
Secondary: 60G17 , 60G42

Keywords: martingale embedding , Skorokhod embedding at a rate , Weighted empirical and finite sampling processes

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.19 • No. 2 • April, 1991
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