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

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.