A generalized Faulhaber inequality, improved bracketing covers, and applications to discrepancy

Math. Comput. Model. Pub Date : 2020-10-22 DOI:10.1090/mcom/3666

M. Gnewuch, Hendrik Pasing, Christian Weiss

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引用次数: 8

Abstract

We prove a generalized Faulhaber inequality to bound the sums of the j j -th powers of the first n n (possibly shifted) natural numbers. With the help of this inequality we are able to improve the known bounds for bracketing numbers of d d -dimensional axis-parallel boxes anchored in 0 0 (or, put differently, of lower left orthants intersected with the d d -dimensional unit cube [ 0 , 1 ] d [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.

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一个广义的Faulhaber不等式，改进的括号覆盖，以及对差异的应用

我们证明了一个广义的Faulhaber不等式来约束前n n个(可能移位的)自然数的j - j次幂的和。在这个不等式的帮助下，我们能够改进锚定在0 0(或者换句话说，与d d维单位立方体[0,1]d [0,1]^d相交的左下邻边)的d d维轴平行盒的括号数的已知界限。我们使用这些改进的括号数建立了负相关随机点集的星差及其期望的新界限。我们也将我们的发现应用于加权星差。

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Math. Comput. Model.

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