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

In this paper 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. Furthermore, we 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.Comment: 29 pages. Second version: some minor changes, typos fixed, etc. The manuscript has been accepted for publication by Acta Arithmetic

Quasi-Monte Carlo algorithms for unbounded, weighted integration problems

In this article we investigate Quasi-Monte Carlo methods for multidimensional improper integrals with respect to a measure other than the uniform distribution. Additionally, the integrand is allowed to be unbounded at the lower boundary of the integration domain. We establish convergence of the Quasi-Monte Carlo estimator to the value of the improper integral under conditions involving both the integrand and the sequence used. Furthermore, we suggest a modification of an approach proposed by Hlawka and Mück for the creation of low-discrepancy sequences with regard to a given density, which are suited for singular integrands. Key words: Quasi-Monte Carlo integration, weighted integration, non-uniformly distributed low-discrepancy sequences This paper is devoted to Quasi-Monte Carlo (QMC) techniques for weighted integration problems of the for