In this work, we analyze the convergence rate of randomized quasi-Monte Carlo (RQMC) methods under Owen's boundary growth condition [Owen, 2006] via spectral analysis. Specifically, we examine the RQMC estimator variance for the two commonly studied sequences: the lattice rule and the Sobol' sequence, applying the Fourier transform and Walsh--Fourier transform, respectively, for this analysis. Assuming certain regularity conditions, our findings reveal that the asymptotic convergence rate of the RQMC estimator's variance closely aligns with the exponent specified in Owen's boundary growth condition for both sequence types. We also provide guidance on choosing the importance sampling density to minimize RQMC estimator variance.

翻译：本文通过谱分析方法，研究了Owen边界增长条件[Owen, 2006]下随机化拟蒙特卡洛（RQMC）方法的收敛速率。具体而言，我们分别利用傅里叶变换和沃尔什-傅里叶变换，分析了两种常用序列（格子规则与Sobol'序列）的RQMC估计量方差。在特定正则性假设下，我们的结果表明：对于这两种序列类型，RQMC估计量方差的渐近收敛速率与Owen边界增长条件中指定的指数高度吻合。此外，我们还提供了选择重要性抽样密度以最小化RQMC估计量方差的指导原则。