Affiliation:
1. University of Washington , Department of Mathematics , Seattle , Washington, U.S.A
2. Ruhr West University of Applied Sciences , Department of Natural Sciences , Mülheim an der Ruhr , Germany
Abstract
Abstract
We provide an algorithm to approximate a finitely supported discrete measure μ by a measure νN
corresponding to a set of N points so that the total variation between μ and νN
has an upper bound. As a consequence if μ is a (finite or infinitely supported) discrete probability measure on [0, 1]
d
with a sufficient decay rate on the weights of each point, then μ can be approximated by νN
with total variation, and hence star-discrepancy, bounded above by (log N)N−
1. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure μ, there exist finite sets whose star-discrepancy with respect to μ is at most
(
log
N
)
d
−
1
2
N
−
1
{\left( {\log \,N} \right)^{d - {1 \over 2}}}{N^{ - 1}}
. Moreover, we close a gap in the literature for discrepancy in the case d =1 showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure.
Reference18 articles.
1. [1] AISTLEITNER, C.—BILYK, D.—NIKOLOV, A.: Tusnády’s problem, the transference principle, and non-uniform QMC sampling. In: Monte Carlo and Quasi-Monte Carlo Methods—MCQMC 2016, Springer Proc. Math. Stat. Vol. 241, Springer, Cham, 2018 pp. 169–180.
2. [2] AISTLEITNER, C.—DICK, J.: Low-discrepancy point sets for non-uniform measures, Acta Arith. 163 (4) (2014), 345–369.10.4064/aa163-4-4
3. [3] ABRAMOWITZ, M.—STEGUN, I. A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, Vol. 55. U.S. Government Printing Office, Washington, D. C. 1948.
4. [4] BECK, J.: Some upper bounds in the theory of irregularities of distribution, Acta Arith. 43 (2) (1984), 115–130.10.4064/aa-43-2-115-130
5. [5] CARBONE, I.: Discrepancy of LS-sequences of partitions and points, Ann. Mat. Pura Appl. 191 (2012), 819–844.10.1007/s10231-011-0208-z
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献