Randomized quasi-Monte Carlo and Owen's boundary growth condition: A spectral analysis

TL;DR

This paper investigates the convergence rates of randomized quasi-Monte Carlo methods under Owen's boundary growth condition using spectral analysis, focusing on lattice and Sobol' sequences and their variance decay.

Contribution

It provides a spectral analysis linking Owen's boundary growth condition to the asymptotic convergence rate of RQMC estimators for key sequences.

Findings

Convergence rate aligns with Owen's boundary growth exponent.

Spectral analysis applies Fourier and Walsh transforms.

Results extend to certain discontinuous integrands.

Abstract

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.