A GENERALIZED FAULHABER INEQUALITY, IMPROVED BRACKETING COVERS, AND APPLICATIONS TO DISCREPANCY
| Autor(en): | Gnewuch, Michael Pasing, Hendrik Weiss, Christian |
Stichwörter: | ALGORITHMS; BOUNDS; bracketing number; covering number; Faulhaber's formula; INVERSE; Mathematics; Mathematics, Applied; Monte Carlo point sets; negative correlation; NEGATIVE DEPENDENCE; NUMBERS; pre-asymptotic bound; SMALL BALL INEQUALITY; STAR-DISCREPANCY; sums of powers; tractability; weighted star-discrepancy | Erscheinungsdatum: | 2021 | Herausgeber: | AMER MATHEMATICAL SOC | Journal: | MATHEMATICS OF COMPUTATION | Volumen: | 90 | Ausgabe: | 332 | Startseite: | 2873 | Seitenende: | 2898 | Zusammenfassung: | We prove a generalized Faulhaber inequality to bound the sums of the j-th powers of the first n (possibly shifted) natural numbers. With the help of this inequality we are able to improve the known bounds for bracketing numbers of d-dimensional axis-parallel boxes anchored in 0 (or, put differently, of lower left orthants intersected with the d-dimensional unit cube [0, 1](d)). We use these improved bracketing numbers to establish new bounds for the star-discrepancy of negatively dependent random point sets and its expectation. We apply our findings also to the weighted star-discrepancy. |
ISSN: | 00255718 | DOI: | 10.1090/mcom/3666 |
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