On Monday, October 14th, at 2pm, in Salle Poincaré, Michael Gnewuch (U. Osnabrück) will give a talk entitled: On Negative Dependence Properties of Sampling Schemes and Probabilistic Discrepancy Bounds.
Abstract: In several problems as, e.g., numerical integration or one-shot optimization, one is interested in having sample points that are uniformly distributed and have certain additional nice features, as, e.g, very regular low-dimensional projections. As a measure of uniformity of distribution we consider the star discrepancy. Especially in high dimension, Monte Carlo points exhibit good uniform distribution properties, but will usually not exhibit the additional nice features one is interested in. In this talk we present a relaxed notion of negative orthant dependence that still allows for large deviation bounds. We present a theorem stating that sample points that satisfy this notion have a small star discrepancy that is at least as good (or only slightly worse) than the one of Monte Carlo points. The hope is that this class of sample point sets contains sets that have the desired additional features.The notion is, for instance, satisfied by Latin hypercube samples. We show that a relaxation of the notion of negative orthant dependence is essential, otherwise the theorem cannot be applied to Latin hypercube sampling or to so-called (t, m, s)-nets.