For $m, d \in {\mathbb N}$, a jittered sampling point set $P$ having $N = m^d$ points in $[0,1)^d$ is constructed by partitioning the unit cube $[0,1)^d$ into $m^d$ axis-aligned cubes of equal size and then placing one point independently and uniformly at random in each cube. We show that there are constants $c \ge 0$ and $C$ such that for all $d$ and all $m \ge d$ the expected non-normalized star discrepancy of a jittered sampling point set satisfies \[c \,dm^{\frac{d-1}{2}} \sqrt{1 + \log(\tfrac md)} \le {\mathbb E} D^*(P) \le C\, dm^{\frac{d-1}{2}} \sqrt{1 + \log(\tfrac md)}.\] This discrepancy is thus smaller by a factor of $\Theta\big(\sqrt{\frac{1+\log(m/d)}{m/d}}\,\big)$ than the one of a uniformly distributed random point set of $m^d$ points. This result improves both the upper and the lower bound for the discrepancy of jittered sampling given by Pausinger and Steinerberger (Journal of Complexity (2016)). It also removes the asymptotic requirement that $m$ is sufficiently large compared to $d$.

中文翻译：

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抖动采样的急剧差异范围

对于$ m，d \ in {\ mathbb N} $，通过对单位立方体$ [分区]构造一个抖动采样点集$ P $，其中$ N = $ [0,1）^ d $中的m ^ d $个点。将0,1）^ d $放入大小相等的$ m ^ d $轴对齐的多维数据集中，然后在每个多维数据集中独立且均匀地随机放置一个点。我们证明存在常数$ c \ ge 0 $和$ C $使得对于所有$ d $和所有$ m \ ge d $，抖动采样点集的预期非标准化星形差异都满足\ [c \， dm ^ {\ frac {d-1} {2}} \ sqrt {1 + \ log（\ tfrac md）} \ le {\ mathbb E} D ^ *（P）\ le C \，dm ^ {\ frac {d-1} {2}} \ sqrt {1 + \ log（\ tfrac md）}。\]因此，此差异减小了$ \ Theta \ big（\ sqrt {\ frac {1+ \ log （m / d）} {m / d}} \，\ big）$，而不是$ m ^ d $点的均匀分布随机点集合中的一个。此结果改善了Pausinger和Steinerberger给出的抖动采样差异的上限和下限（复杂性杂志（2016））。它还消除了渐进性要求，即$ m $与$ d $相比足够大。