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.

翻译：设 $f:[0,1]^d\to\mathbb{R}$ 为 Aistleitner 和 Dick (2015) 定义的完全单调被积函数，且点集 $\boldsymbol{x}_0,\dots,\boldsymbol{x}_{n-1}\in[0,1]^d$ 在 $[0,1]^d$ 中处处具有非负局部偏差（NNLD）。我们展示了如何利用这些性质得到 $f$ 在 $[0,1]^d$ 上积分的非渐近可计算上界。类似的非正局部偏差（NPLD）性质可提供可计算下界。自 Gabai (1967) 以来已知，任意基数 $b\ge2$ 的二维 Hammersley 点具有非负局部偏差。利用关联随机变量的概率概念，我们将 Gabai 的发现推广到任意基数 $b\ge2$ 和任意维度 $d\ge1$ 的数字化网络，当生成矩阵为置换矩阵时成立。我们证明：当 $d\ge3$ 时，置换矩阵无法达到数字网络质量参数的最佳值。因此，我们提供的可计算绝对可靠界在高维情形下的估计精度低于常规数字网络估计。此外，我们还能构造具有 NNLD 性质的高维秩一格点规则，并证明这些格点不具备良好的偏差性质：任何满足 NNLD 性质的 $d\ge2$ 维格点规则要么不是投影正则的，要么其所有点均位于主对角线上。