AI 中文总结
该研究分析了不同随机化过程对拟蒙特卡罗点集局部几何的影响，明确了各类抽样及加扰方法的网格比概率阶，揭示随机化几何效应取决于是否引入局部独立性或共享代数随机性。
AI 中文摘要
我们研究标准随机化过程如何影响拟蒙特卡罗点集的局部几何，以其最小距离和网格比衡量。尽管结构化格族内的概率选择可生成拟均匀点集，但对现有低偏差构造进行随机化未必能保持拟均匀性。我们首先确定蒙特卡罗、抖动和拉丁超立方抽样的精确概率阶，其网格比随N的正幂发散：蒙特卡罗抽样为Θ_P(N^(1/d)(log N)^(1/d))，抖动抽样为Θ_P(N^(1/(d+1)))，d≥2时拉丁超立方抽样为Θ_P(N^(1/d)(log N)^(1/d))。我们还得到抖动样本最小距离的威布尔极限律。对于完全Owen加扰，每个固定t的网族最小距离为O_P(N^(-3/(2d)))，因此其网格比为Ω_P(N^(1/(2d)))。在均匀重合和公共前缀条件下，这些界在对数因子内是精确的。此外，任何(t,d)-序列的单次完全Owen加扰几乎必然非拟均匀。相比之下，对于d≥2维中固定t的二进制数字网的矩阵和线性加扰，其网格比为O_P(log N)，而在独立平衡前缀仿射尾模型中为Θ_P(log N)。该模型还给出一维二进制数字(0,m,1)-网在矩阵或线性加扰下的精确概率阶。这些结果表明，随机化的几何效应由其是否引入局部独立性或共享代数随机性决定。
英文摘要
We study how standard randomization procedures affect the local geometry of quasi-Monte Carlo point sets, as measured by their minimum distance and mesh ratio. Although probabilistic selection within structured lattice families can produce quasi-uniform point sets, randomizing an existing low-discrepancy construction need not preserve quasi-uniformity. We first determine sharp probabilistic orders for Monte Carlo, jittered, and Latin hypercube sampling, whose mesh ratios diverge as positive powers of $N$. The orders are $Θ_{\mathbb{P}}(N^{1/d}(\log N)^{1/d})$ for Monte Carlo sampling, $Θ_{\mathbb{P}}(N^{1/(d+1)})$ for jittered sampling, and, for $d\ge 2$, $Θ_{\mathbb{P}}(N^{1/d}(\log N)^{1/d})$ for Latin hypercube sampling. We also obtain a Weibull limit law for the minimum distance of jittered samples. For full Owen scrambling, every family of fixed-$t$ nets has minimum distance $O_{\mathbb{P}}(N^{-3/(2d)})$, and its mesh ratio is therefore $Ω_{\mathbb{P}}(N^{1/(2d)})$. Under uniform coincidence and common-prefix conditions, these bounds are sharp up to logarithmic factors. Moreover, a single full Owen scrambling of any $(t,d)$-sequence is almost surely non-quasi-uniform. By contrast, for matrix and linear scrambling of binary digital nets with fixed $t$ in dimension $d\ge 2$, the mesh ratio is $O_{\mathbb{P}}(\log N)$, whereas it is $Θ_{\mathbb{P}}(\log N)$ in the separate balanced-prefix affine-tail model. The model also yields the exact probabilistic order for one-dimensional binary digital $(0,m,1)$-nets under matrix or linear scrambling. These results demonstrate that the geometric effect of randomization is governed by whether it introduces local independence or shared algebraic randomness.
