The hybrid spectral test

TL;DR

This paper introduces a hybrid spectral test combining trigonometric, Walsh, and b-adic functions to analyze the uniform distribution of digital sequences, enhancing tools for Quasi-Monte Carlo methods and pseudo-random number generators.

Contribution

It develops the b-adic method and hybrid spectral test, integrating structural properties of b-adic integers for improved analysis of digital sequences' uniformity.

Findings

The hybrid spectral test effectively measures uniform distribution.

Discrepancy can be closely approximated by the spectral test.

The method applies to Quasi-Monte Carlo and pseudo-random number generators.

Abstract

The starting point of this paper is the interplay between the construction principle of a sequence and the characters of the compact abelian group that underlies the construction. In case of the Halton sequence in base $\mathbf b=(b_1, \ldots, b_s)$ in the $s$-dimensional unit cube $[0,1)^s$, which is an important type of a digital sequence, this kind of duality principle leads to the so-called $\mathbf b$-adic function system and provides the basis for the $\mathbf b$-adic method, which we present in connection with hybrid sequences. This method employs structural properties of the compact group of $\mathbf b$-adic integers as well as $\mathbf b$-adic arithmetic to derive tools for the analysis of the uniform distribution of sequences in $[0,1)^s$. We first clarify the point which function systems are needed to analyze digital sequences. Then, we present the hybrid spectral test in terms of trigonometric-, Walsh-, and $\mathbf b$-adic functions. Various notions of diaphony as well as many figures of merit for rank-1 quadrature rules in Quasi-Monte Carlo integration and for certain linear types of pseudo-random number generators are included in this measure of uniform distribution. Further, discrepancy may be approximated arbitrarily close by suitable versions of the spectral test.