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Random and Quasi-Random Point Sets

  • Peter Hellekalek
  • Gerhard Larcher

Part of the Lecture Notes in Statistics book series (LNS, volume 138)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Fred J. Hickernell
    Pages 109-166
  3. Gerhard Larcher
    Pages 167-222
  4. Pierre L’Ecuyer, Peter Hellekalek
    Pages 223-265
  5. Harald Niederreiter, Chaoping Xing
    Pages 267-302
  6. Back Matter
    Pages 333-334

About this book

Introduction

This volume is a collection of survey papers on recent developments in the fields of quasi-Monte Carlo methods and uniform random number generation. We will cover a broad spectrum of questions, from advanced metric number theory to pricing financial derivatives. The Monte Carlo method is one of the most important tools of system modeling. Deterministic algorithms, so-called uniform random number gen­ erators, are used to produce the input for the model systems on computers. Such generators are assessed by theoretical ("a priori") and by empirical tests. In the a priori analysis, we study figures of merit that measure the uniformity of certain high-dimensional "random" point sets. The degree of uniformity is strongly related to the degree of correlations within the random numbers. The quasi-Monte Carlo approach aims at improving the rate of conver­ gence in the Monte Carlo method by number-theoretic techniques. It yields deterministic bounds for the approximation error. The main mathematical tool here are so-called low-discrepancy sequences. These "quasi-random" points are produced by deterministic algorithms and should be as "super"­ uniformly distributed as possible. Hence, both in uniform random number generation and in quasi-Monte Carlo methods, we study the uniformity of deterministically generated point sets in high dimensions. By a (common) abuse oflanguage, one speaks of random and quasi-random point sets. The central questions treated in this book are (i) how to generate, (ii) how to analyze, and (iii) how to apply such high-dimensional point sets.

Keywords

Generator Kernel LDA Probability theory statistics

Editors and affiliations

  • Peter Hellekalek
    • 1
  • Gerhard Larcher
    • 1
  1. 1.Institut für MathematikUniversität SalzburgSalzburgAustria

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-1702-2
  • Copyright Information Springer-Verlag New York, Inc. 1998
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-98554-1
  • Online ISBN 978-1-4612-1702-2
  • Series Print ISSN 0930-0325
  • Buy this book on publisher's site