Low-discrepancy point sets for non-uniform measures

• Autorzy: Christoph Aistleitner , Josef Dick
• Języki publikacji: EN

Abstrakty

EN

We prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure μ on the d-dimensional unit cube. We improve a theorem of Beck, by showing that for any d ≥ 1, N ≥ 1, and any non-negative, normalized Borel measure μ on $[0,1]^d$ there exists a point set $x_1, ..., x_N ∈ [0,1]^d$ whose star-discrepancy with respect to μ is of order
$D_N*(x_1, ..., x_N; μ ) ≪ ((log N)^{(3d+1)/2})/N$.
For the proof we use a theorem of Banaszczyk concerning the balancing of vectors, which implies an upper bound for the linear discrepancy of hypergraphs. Furthermore, the theory of large deviation bounds for empirical processes indexed by sets is discussed, and we prove a numerically explicit upper bound for the inverse of the discrepancy for Vapnik-Chervonenkis classes. Finally, using a recent version of the Koksma-Hlawka inequality due to Brandolini, Colzani, Gigante and Travaglini, we show that our results imply the existence of cubature rules yielding fast convergence rates for the numerical integration of functions having discontinuities of a certain form.

Kategorie tematyczne

• 62G30: Order statistics; empirical distribution functions
• 11K38: Irregularities of distribution, discrepancy
• 65C05: Monte Carlo methods
• 65D30: Numerical integration

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 163
• Numer: 4
• Strony: 345-369

Daty

wydano

2014

Twórcy

autor

Christoph Aistleitner

• School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

autor

Josef Dick

• School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

Identyfikatory

DOI

10.4064/aa163-4-4