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We extend the notion of jittered sampling to arbitrary partitions and studythe discrepancy of the related point sets. Let$\mathbf{\Omega}=(\Omega_1,\ldots,\Omega_N)$ be a partition of $[0,1]^d$ andlet the $i$th point in $\mathcal{P}$ be chosen uniformly in the $i$th set ofthe partition (and stochastically independent of the other points),$i=1,\ldots,N$. For the study of such sets we introduce the concept of auniformly distributed triangular array and compare this notion to relatednotions 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 equivolumepartitions of the unit cube into convex sets or into sets with a uniformpositive lower bound on their reach. It is shown that these classes contain atleast one minimizer of the expected ${\mathcal{L}_p}$-discrepancy. Weillustrate our results with explicit constructions for small $N$. In addition,we present a family of partitions that seems to improve the expecteddiscrepancy of Monte Carlo sampling by a factor of 2 for every $N$.

Discrepancy of stratified samples from partitions of the unit cube