MATHEMATICS OF COMPUTATION Volume 73, Number 248, Pages 1903–1911 S 0025-5718(04)01653-9 Article electronically published on April 22, 2004 ON STRONG TRACTABILITY OF WEIGHTED MULTIVARIATE INTEGRATION FRED J. HICKERNELL, IAN H. SLOAN, AND GRZEGORZ W. WASILKOWSKI Abstract. We prove that for every dimension s and every number n of points, there exists a point-set Pn,s whose -weighted unanchored L∞ discrepancy is bounded from above by C(b)/n 1/2-b independently of s provided that the sequence = {γ k } has ∑ ∞ k=1 γ a k < ∞ for some (even arbitrarily large) a. Here b is a positive number that could be chosen arbitrarily close to zero and C(b) depends on b but not on s or n. This result yields strong tractability of the corresponding integration problems including approximation of weighted integrals R D f (x) ρ(x) dx over unbounded domains such as D = R s . It also supplements the results that provide an upper bound of the form C p s/n when γ k ≡ 1. 1. Introduction This article studies the unanchored L ∞ discrepancy and strong tractability of the corresponding integration problem. We begin the discussion with the integration problem. Consider approximating the following type of weighted integrals: (1) I ρ (f )= Z D f (x) ρ(x) dx. Here D is an s-dimensional box, (2) D = (a 1 ,b 1 ) ×···× (a s ,b s ) ⊆ R s , with possibly infinite a i and/or b i . It is assumed that the weight function ρ has a tensor product form, (3) ρ(x)= s Y k=1 ρ k (x k ), for nonnegative and Lebesgue integrable functions ρ k . For simplicity, it is assumed that the ρ k are probability densities on (a k ,b k ), i.e., Z b k a k ρ k (x) dx =1. However, as explained in Section 6.1 in [3], it is sufficient to assume that the integrals of ρ k are finite. Received by the editor December 16, 2002 and, in revised form, April 30, 2003. 2000 Mathematics Subject Classification. Primary 65D30, 65D32, 65Y20, 11K38. Key words and phrases. Weighted integration, quasi–Monte Carlo methods, low discrepancy points, tractability. c 2004 American Mathematical Society 1903 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use