Integrability of weak mixed first-order derivatives and convergence rates of scrambled digital nets

TL;DR

This paper investigates the relationship between the integrability of weak mixed derivatives of functions and the convergence rates of scrambled digital nets, providing theoretical bounds and numerical validation.

Contribution

It establishes a link between the $L^p$ integrability of weak mixed derivatives and the boundedness of generalized Vitali variation, leading to new convergence rate results for scrambled digital nets.

Findings

Bounded generalized Vitali variation by $L^p$ norm of derivatives

Variance convergence rate of scrambled digital nets established

Numerical experiments confirm theoretical predictions

Abstract

We consider the $L^p$ integrability of weak mixed first-order derivatives of the integrand and study convergence rates of scrambled digital nets. We show that the generalized Vitali variation with parameter $\alpha \in [\frac{1}{2}, 1]$ from [Dick and Pillichshammer, 2010] is bounded above by the $L^p$ norm of the weak mixed first-order derivative, where $p = \frac{2}{3-2\alpha}$. Consequently, when the weak mixed first-order derivative belongs to $L^p$ for $1 \leq p \leq 2$, the variance of the scrambled digital nets estimator convergences at a rate of $\mathcal{O}(N^{-4+\frac{2}{p}} \log^{s-1} N)$. Numerical experiments further validate the theoretical results.