Discrepancy of stratified samples from partitions of the unit cube

Abstract: We extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let $\mathbf{\Omega}=(\Omega_1,\ldots,\Omega_N)$ be a partition of $[0,1]^d$ and let the $i$th point in $\mathcal{P}$ be chosen uniformly in the $i$th set of the partition (and stochastically independent of the other points), $i=1,\ldots,N$. For the study of such sets we introduce the concept of a uniformly distributed triangular array and compare this notion to related notions in the literature. We prove that the expected ${\mathcal{L}p}$-discrepancy, $\mathbb{E} {\mathcal{L}_p}(\mathcal{P}{\mathbf{\Omega}})^p$, of a point set $\mathcal{P}_\mathbf{\Omega}$ generated from any equivolume partition $\mathbf{\Omega}$ is always strictly smaller than the expected ${\mathcal{L}_p}$-discrepancy of a set of $N$ uniform random samples for $p>1$. For fixed $N$ we consider classes of stratified samples based on equivolume partitions of the unit cube into convex sets or into sets with a uniform positive lower bound on their reach. It is shown that these classes contain at least one minimizer of the expected ${\mathcal{L}_p}$-discrepancy. We illustrate our results with explicit constructions for small $N$. In addition, we present a family of partitions that seems to improve the expected discrepancy of Monte Carlo sampling by a factor of 2 for every $N$.