Recent advances in quasi-Monte Carlo integration have shown that for linearly scrambled digital net estimators, the convergence rate can be dramatically improved by taking the median rather than the mean of multiple independent replicates. In this work, we demonstrate that the quantiles of such estimators can be used to construct confidence intervals with asymptotically valid coverage for high-dimensional integrals. By analyzing the error distribution for a class of infinitely differentiable integrands, we prove that as the sample size increases, the integration error decomposes into an asymptotically symmetric component and a vanishing remainder. Consequently, the asymptotic error distribution is symmetric about zero, ensuring that a quantile-based interval constructed from independent replicates captures the true integral with probability converging to a nominal level determined by the binomial distribution.

翻译：拟蒙特卡洛积分的最新进展表明，对于线性扰动的数字网络估计量，通过取多个独立重复样本的中位数而非均值，可以显著提高收敛速度。本文证明，此类估计量的分位数可用于构建高维积分置信区间，该区间具有渐近有效的覆盖概率。通过分析一类无限可微被积函数的误差分布，我们证明随着样本量的增加，积分误差可分解为一个渐近对称的分量和一个趋于零的余项。因此，渐近误差分布关于零对称，这保证了基于独立重复样本构建的分位数区间能以概率收敛至由二项分布确定的标称水平来捕获真实积分值。