Functions of bounded variation, signed measures, and a general Koksma–Hlawka inequality

Christoph Aistleitner¹ ; Josef Dick²

¹ Department of Mathematics and Statistics Graduate School of Science Kobe University 1-1 Rokkodai, Nada-ku Kobe 657-8501, Japan
² School of Mathematics and Statistics University of New South Wales Sydney, NSW 2052, Australia

Acta Arithmetica, Tome 167 (2015) no. 2, pp. 143-171

Résumé

We prove a correspondence principle between multivariate functions of bounded variation in the sense of Hardy and Krause and signed measures of finite total variation, which allows us to obtain a simple proof of a generalized Koksma–Hlawka inequality for non-uniform measures. Applications of this inequality to importance sampling in Quasi-Monte Carlo integration and tractability theory are given. We also discuss the problem of transforming a low-discrepancy sequence with respect to the uniform measure into a sequence with low discrepancy with respect to a general measure $\mu $, and show the limitations of a method suggested by Chelson.

Détail

DOI : 10.4064/aa167-2-4

Keywords: prove correspondence principle between multivariate functions bounded variation sense hardy krause signed measures finite total variation which allows obtain simple proof generalized koksma hlawka inequality non uniform measures applications inequality importance sampling quasi monte carlo integration tractability theory given discuss problem transforming low discrepancy sequence respect uniform measure sequence low discrepancy respect general measure limitations method suggested chelson