A refinement of the KMT inequality for the uniform empirical process. (English) Zbl 0638.60040

Mason, David M.; van Zwet, Willem R.

Ann. Probab. 15, 871-884 (1987).

The authors give the first detailed proof of the strong approximation up to the error \(O(n^{-}) \log n\) of the uniform empirical process by a Brownian bridge process which had been outlined in J. Komlós, P. Major and G. Tusnady, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 34, 33-58 (1976; Zbl 0307.60045). They sharpen this result by providing a strong approximation result for the differences of these processes weighted to variance one, extending earlier work on weighted approximations. The proof of these important results relies on several sharp inequalities which may be of independent interest.

Reviewer: F.Götze

MSC:

60F15	Strong limit theorems
60G17	Sample path properties

Keywords:

uniform empirical process; KMT inequality; strong approximation; Brownian bridge process; weighted approximations; sharp inequalities

Citations:

Zbl 0315.60031; Zbl 0307.60045