Discrepancy between Qmc and Rqmc

We introduce a class of functions in d ≥ 3 dimensions which have arbitrary odd superposition effective dimensions between three and d inclusive. We prove that for the integration of any function in this class any Sobol’ points of a fixed length have zero error, whereas Owen’s scrambling of any Sobol’ points of the same length has the same variance of error as simple Monte Carlo methods. Furthermore, for any function in the same class Owen’s scrambling of highdiscrepancy points, which consist of d copies of the van der Corput points in base two, gives zero-variance estimates for the integration. Communicated by Reinhard Winkler Dedicated to Professor Robert F. Tichy on the occasion of his 50th birthday