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 analysis for certain discontinuous integrands.

翻译：本文通过谱分析方法，研究了在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，我们针对两种常用序列——格点规则与Sobol'序列，分别应用傅里叶变换与沃尔什-傅里叶变换，分析了RQMC估计量的方差。在特定正则性假设下，研究结果表明：对于两类序列，RQMC估计量方差的渐近收敛速率均与Owen边界增长条件中指定的指数高度吻合。本文还对某些不连续被积函数进行了相应分析。