Randomized quasi-Monte Carlo and Owen’s boundary growth condition: a spectral analysis

Abstract In this work we analyse the convergence rate of randomized quasi-Monte Carlo (RQMC) methods under Owen’s boundary growth condition (Owen, A. B. (2006) Halton sequences avoid the origin. SIAM Rev., 48, 487−503.) via spectral analysis. Specifically, we examine the RQMC estimator variance for the two commonly studied sequences: the lattice rule and the Sobol’ sequence. We apply the Fourier transform and Walsh–Fourier transform, respectively. 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.