The Halton sequence and its discrepancy in the Cantor expansion

Alena Haddley,Poj Lertchoosakul,Radhakrishnan Nair

doi:10.1007/s10998-016-0169-5

Alena Haddley, Poj Lertchoosakul + Show 1 more

Open Access

https://doi.org/10.1007/s10998-016-0169-5

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Abstract

We consider variants of the Halton sequence in a generalized numeration system, called the Cantor expansion. We show that it provides a wealth of low-discrepancy sequences by giving an estimate of the (star) discrepancy of the Halton sequence in each bounded Cantor base. The techniques used in our estimation of the discrepancy are adapted from those developed by E.I. Atanassov.

Highlights

Let ω =∞ n=1 be a sequence in [0, 1)s
A standard problem in numerical analysis is estimating the integral of a function, through a knowledge of its value at a finite number of points of the sequence. This is known as the Monte Carlo method in the case of stochastic sequencesnN=1 or the quasi-Monte Carlo method in the case of deterministicnN=1
We introduce the Halton sequence in a generalized numeration system, which is induced by the a-adic integers and which is called the Cantor expansion, and give an estimate of its discrepancy by adapting the techniques developed by Atanassov

Summary

Introduction

We introduce the Halton sequence in a generalized numeration system, which is induced by the a-adic integers and which is called the Cantor expansion, and give an estimate of its discrepancy by adapting the techniques developed by Atanassov. It is worth noting here that the van der Corput sequence and some one-dimensional low-discrepancy sequences with respect to the Cantor expansion were studied in [2] and [5] Note that it was mentioned in [8] about the Halton sequence in a more generalized numeration system than the Cantor expansion, called the G-expansion; the paper aimed to study the Halton sequence in some fixed non-integer bases and did not touch on the Halton sequence with respect to dynamical bases. 5, we prove an estimate of discrepancy of the van der Corput sequence in a generalized numeration system without the restriction on boundedness of inducing sequences.

A generalized Halton sequence

Preliminary lemmas

Proof of the main theorem

An open problem

A generalized Hammersley point set