Constructing a new class of low-discrepancy sequences by using the β-adic transformation

Abstract

It is well known that low-discrepancy sequences and their discrepancy play essential roles in quasi Monte Carlo methods [5]. In this paper, a new class of low-discrepancy sequences Nβ is constructed by using the ergodic theoretical transformation which is called β-adic transformation [7, 8]. Here, β is a real number greater than 1. When β is an integer greater than 2, Nβ becomes the classical van der Corput sequence in base β. Therefore, the class Nβ can be regarded as a generalization of the van der Corput sequence. It is shown that for some special β, the discrepancy of this sequence decreases in the fastest order O(N−1logN). We give the numerical results of discrepancy of Nβ for some βs. Pagès [6] also generalized van der Corput sequence in a different direction by using an ergodic transformation.