Quasi-Monte Carlo (QMC) methods are equal weight quadrature rules to approximate integrals over the unit cube with respect to the uniform measure. In this paper we discuss QMC integration with respect to general product measures defined on an arbitrary cube. We only require that the cumulative distribution function is invertible. We develop a worst-case error bound and study the dependence of the error on the number of points and the dimension for digital nets and sequences as well as polynomial lattice point sets, which are mapped to the domain using the inverse cumulative distribution function. We do not require any smoothness properties of the probability density function and the worst-case error does not depend on the particular choice of density function and its smoothness. The component-by-component construction of polynomial lattice rules is based on a criterion which depends only on the size of the cube but is otherwise independent of the product measure.

中文翻译：

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基于数字网络和序列的立方体加权积分

准蒙特卡罗 (QMC) 方法是等权重求积规则，用于近似单位立方体上关于统一测度的积分。在本文中，我们讨论 QMC 集成与定义在任意立方体上的一般产品度量。我们只要求累积分布函数是可逆的。我们开发了最坏情况的误差界限，并研究了误差对数字网络和序列以及多项式格点集的点数和维度的依赖性，这些点集使用逆累积分布函数映射到域。我们不需要概率密度函数的任何平滑特性，最坏情况下的误差不依赖于密度函数的特定选择及其平滑度。