Improved bounds on the gain coefficients for digital nets in prime power base

TL;DR

This paper derives improved upper bounds on gain coefficients for digital nets in prime power bases, enhancing understanding of their variance decay properties in randomized quasi-Monte Carlo integration.

Contribution

It provides a unified framework to bound gain coefficients for digital nets in prime power bases, generalizing previous results and explaining known bounds.

Findings

Improved upper bounds on gain coefficients for digital nets in prime power bases.

Unified explanation of previous bounds for Faure and Sobol' sequences.

Nonzero gain coefficients in base 2 are powers of two.

Abstract

We study randomized quasi-Monte Carlo integration by scrambled nets. The scrambled net quadrature has long gained its popularity because it is an unbiased estimator of the true integral, allows for a practical error estimation, achieves a high order decay of the variance for smooth functions, and works even for $L^p$-functions with any $p\geq 1$. The variance of the scrambled net quadrature for $L^2$-functions can be evaluated through the set of the so-called gain coefficients. In this paper, based on the system of Walsh functions and the concept of dual nets, we provide improved upper bounds on the gain coefficients for digital nets in general prime power base. Our results explain the known bound by Owen (1997) for Faure sequences, the recently improved bound by Pan and Owen (2021) for digital nets in base 2 (including Sobol' sequences as a special case), and their finding that all the nonzero gain coefficients for digital nets in base 2 must be powers of two, all in a unified way.