An inequality for differences of distribution functions
type: Item
instance of: scholarly article
title: An inequality for differences of distribution functions
zbMATH Open document ID: 0697.10041
DOI: 10.1007/BF01198113
author: Harald Niederreiter
author: Robert F. Tichy
author: Gerhard Turnwald
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: Let F(x), G(x) be distribution functions on \([0,1]^ s\), \(D=F(x)-G(x)\). Then the \(L^ p\)-norm \(\| D\|_ p\) is trivially bounded by the supremum norm \(\| D\|_{\infty}\). The authors show that, under mild regularity conditions, \(\| D\|_{\infty}\) can also be bounded in terms of \(\| D\|_ p.\) In reality, the theorem proved by the authors is even more general. An important special case is obtained for \(G=\sum^{N}_{n=1}1_{[0,x)}(x_ n)\), \(x\in [0,1]^ s\). Then \(\| D\|_{\infty}\) is the star-discrepancy of \(x_ 1,x_ 2,...,x_ N\) with respect to F(x). Thus earlier results by \textit{P. D. Projnov} [Proc. Am. Math. Soc. 95, 527-532 (1985; Zbl 0598.10056) and C. R. Acad. Bulg. Sci. 38, 1465-1468 (1985; Zbl 0589.10054)] are considerably generalized.
Mathematics Subject Classification ID: 11K06
zbMATH DE Number: 4142139
zbMATH Keywords: uniform distribution of sequences
zbMATH Keywords: \(L^ p\)-norm
zbMATH Keywords: supremum norm
zbMATH Keywords: star- discrepancy
reviewed by: Peter Schatte
MaRDI profile type: Publication
cites work: Generalization of Two Results of the Theory of Uniform Distribution
cites work: Q3717150
cites work: On Inequalities for Large Deviations for Certain Statistics
full work available at URL: https://doi.org/10.1007/bf01198113
OpenAlex ID: W1998232556
description: scientific article; zbMATH DE number 4142139