AI 中文总结
研究通过扰动特定网格得到的随机采样方法偏差，该方法由概率测度$\mu$参数化。发现除狄拉克测度外，概率测度在蒙特卡罗偏差中类似勒贝格测度，证明$\mu$支撑集直径$\leq c/M$时能达最低偏差且上界精确。
AI 中文摘要
本文研究了通过扰动网格$\frac{1}{M}\mathbb{Z}^{d}\cap\left[ -1/2,1/2\right)^{d}$（其中$M$是一个大的正整数）得到的一族随机采样方法的偏差。该族方法由任意概率测度$\mu$参数化，包含了几种评估$\mathbb{T}^{d}$中$N$点集（$N = M^{d}$）质量的经典方法。我们表明，除狄拉克测度外，所有概率测度在蒙特卡罗偏差中表现得像勒贝格测度。这代表了测度$\mu$依赖于$M$的一种极限情况。在此背景下，我们证明，在直径$\leq c/M$时，$\mu$的支撑集能达到（至多相差一个常数）最低可能的偏差，且此上界是精确的。
英文摘要
This paper investigates the discrepancy of a family of random sampling methods obtained by perturbing the grid $\frac{1}{M}\mathbb{Z}^{d}\cap\left[ -1/2,1/2\right)^{d}$, where $M$ is a large positive integer. Parameterized by an arbitrary probability measure $μ$, this family encompasses several classical methods for evaluating the quality of an $N$-point set in $\mathbb{T}^{d}$, where $N=M^{d}$. We show that all probability measures, except for Dirac measures, behave like the Lebesgue measure in the Monte Carlo discrepancy. This represents a limiting case where the measure $μ$ depends on $M$. In this latter context, we prove that, up to a constant, the lowest possible discrepancy is achieved when the support of $μ$ has diameter $\leq c/M$, and that this upper bound is sharp.
