Let $f:[0,1]^d\to\mathbb{R}$ be a completely monotone integrand as defined by Aistleitner and Dick (2015) and let points $\boldsymbol{x}_0,\dots,\boldsymbol{x}_{n-1}\in[0,1]^d$ have a non-negative local discrepancy (NNLD) everywhere in $[0,1]^d$. We show how to use these properties to get a non-asymptotic and computable upper bound for the integral of $f$ over $[0,1]^d$. An analogous non-positive local discrepancy (NPLD) property provides a computable lower bound. It has been known since Gabai (1967) that the two dimensional Hammersley points in any base $b\ge2$ have non-negative local discrepancy. Using the probabilistic notion of associated random variables, we generalize Gabai's finding to digital nets in any base $b\ge2$ and any dimension $d\ge1$ when the generator matrices are permutation matrices. We show that permutation matrices cannot attain the best values of the digital net quality parameter when $d\ge3$. As a consequence the computable absolutely sure bounds we provide come with less accurate estimates than the usual digital net estimates do in high dimensions. We are also able to construct high dimensional rank one lattice rules that are NNLD. We show that those lattices do not have good discrepancy properties: any lattice rule with the NNLD property in dimension $d\ge2$ either fails to be projection regular or has all its points on the main diagonal. Complete monotonicity is a very strict requirement that for some integrands can be mitigated via a control variate.

翻译：令 $f:[0,1]^d\to\mathbb{R}$ 为 Aistleitner 与 Dick (2015) 所定义的完全单调被积函数，且令点集 $\boldsymbol{x}_0,\dots,\boldsymbol{x}_{n-1}\in[0,1]^d$ 在 $[0,1]^d$ 上处处具有非负局部偏差 (NNLD)。本文展示了如何利用这些性质获得 $f$ 在 $[0,1]^d$ 上积分的非渐近且可计算上界。类似地，非正局部偏差 (NPLD) 性质可提供可计算下界。自 Gabai (1967) 以来已知，任意基数 $b\ge2$ 下的二维 Hammersley 点集具有非负局部偏差。利用随机变量相协性的概率概念，我们将 Gabai 的结论推广至任意基数 $b\ge2$ 和任意维度 $d\ge1$ 下的数字网，其中生成矩阵为置换矩阵。我们证明当 $d\ge3$ 时，置换矩阵无法达到数字网质量参数的最优值。因此，我们提供的可计算绝对确定界在高维情形下的估计精度低于常规数字网估计。我们亦能构造具有 NNLD 性质的高维秩-1格子法则。研究表明此类格点不具备良好的偏差性质：在维度 $d\ge2$ 中，任何具有 NNLD 性质的格子法则要么不满足投影正则性，要么其所有点均位于主对角线上。完全单调性是非常严格的要求，对于某些被积函数可通过控制变量法进行缓解。