The hybrid spectral test

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.