This paper investigates quasi-Monte Carlo (QMC) integration of Lebesgue integrable functions with respect to a density function over $\mathbb{R}^s$. We extend the construction-free median QMC rule to the unanchored weighted Sobolev space of functions defined over $\mathbb{R}^s$. By taking the median of $k=\mathcal{O}(\log N)$ independent randomized QMC estimators, we prove that for any $\epsilon\in(0,r-\frac{1}{2}]$, our method achieves a mean absolute error bound of $\mathcal{O}(N^{-r+\epsilon})$, where $N$ is the number of points and $r>\frac{1}{2}$ is a parameter determined by the function space. This rate matches that of the randomized lattice rules via component-by-component (CBC) construction, while our approach requires no specific CBC constructions or prior knowledge of the space's weight structure. Numerical experiments demonstrate that our method attains accuracy comparable to the CBC method and outperforms the Monte Carlo method.

翻译：本文研究定义在$\mathbb{R}^s$上关于密度函数的Lebesgue可积函数的拟蒙特卡洛（QMC）积分问题。我们将免构造的中位数QMC规则推广至定义在$\mathbb{R}^s$上的无锚加权Sobolev函数空间。通过取$k=\mathcal{O}(\log N)$个独立随机化QMC估计量的中位数，我们证明对于任意$\epsilon\in(0,r-\frac{1}{2}]$，该方法可获得$\mathcal{O}(N^{-r+\epsilon})$阶的平均绝对误差界，其中$N$为点数，$r>\frac{1}{2}$是由函数空间决定的参数。该收敛速率与通过分量逐次（CBC）构造的随机化格点规则一致，而我们的方法无需特定的CBC构造或对空间权重结构的先验知识。数值实验表明，本方法能达到与CBC方法相当的精度，且优于蒙特卡洛方法。