We extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let Ω=(Ω1,…,ΩN)\documentclass[12pt]{minimal}
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\begin{document}$${\varvec{\Omega }}=(\Omega _1,\ldots ,\Omega _N)$$\end{document} be a partition of [0,1]d\documentclass[12pt]{minimal}
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\begin{document}$$[0,1]^d$$\end{document} and let the ith point in P\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {P}}}$$\end{document} be chosen uniformly in the ith set of the partition (and stochastically independent of the other points), i=1,…,N\documentclass[12pt]{minimal}
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\begin{document}$$i=1,\ldots ,N$$\end{document}. 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 Lp\documentclass[12pt]{minimal}
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\begin{document}$${{{\mathcal {L}}}_p}$$\end{document}-discrepancy, ELp(PΩ)p\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {E}}}{{{\mathcal {L}}}_p}({{\mathcal {P}}}_{\varvec{\Omega }})^p$$\end{document}, of a point set PΩ\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {P}}}_{\varvec{\Omega }}$$\end{document} generated from any equivolume partition Ω\documentclass[12pt]{minimal}
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\begin{document}$${\varvec{\Omega }}$$\end{document} is always strictly smaller than the expected Lp\documentclass[12pt]{minimal}
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\begin{document}$${{{\mathcal {L}}}_p}$$\end{document}-discrepancy of a set of N uniform random samples for p>1\documentclass[12pt]{minimal}
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\begin{document}$$p>1$$\end{document}. 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 Lp\documentclass[12pt]{minimal}
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\begin{document}$${{{\mathcal {L}}}_p}$$\end{document}-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.
