An inequality for differences of distribution functions
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.
scientific article; zbMATH DE number 4142139