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

Abstract

Abstract. 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 µ, and show the limitations of a method suggested by Chelson.

Tusnády’s Problem, the Transference Principle, and Non-uniform QMC Sampling

“…A Koksma-Hlawka inequality for general measures was first proved by Götz [17] (see also [2]). For any points x 1 , .…”

On functions of bounded variation

“…However, the proof is far from being trivial (see below). Recall from Lemma 2.1 that a function of bounded HK-variation decomposes into a difference of two completely monotone functions (see also [2]), so the assertion of Theorem 3.1 follows from a similar result for completely monotone functions, stated in Theorem 3.2 below. The fact that completely monotone functions are Borel measurable also seems to be new.…”

Integral Equations, Quasi-Monte Carlo Methods and Risk Modeling

“…For such applications it could help to use a different notion of variation for multivariate functions. Götz [14] proved a version of the Koksma-Hlawka inequality for general measures, Aistleitner & Dick [1] considered functions of bounded variation with respect to signed measures and Brandolini et al [7,6] replaced the integration domain [0, 1] s by an arbitrary bounded Borel subset of R s and proved the inequality for piecewise smooth integrands. Based on fundamental work of Harman [15], a new concept of variation was developed for a wide class of functions, see Pausinger & Svane [25] and Aistleitner et al [2].…”