# Random and Quasi-Random Point Sets

• Peter Hellekalek
• Gerhard Larcher
Book

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

1. Front Matter
Pages i-xii
2. József Beck
Pages 1-48
3. Peter Hellekalek
Pages 49-108
4. Fred J. Hickernell
Pages 109-166
5. Gerhard Larcher
Pages 167-222
6. Pierre L’Ecuyer, Peter Hellekalek
Pages 223-265
7. Harald Niederreiter, Chaoping Xing
Pages 267-302
8. Shu Tezuka
Pages 303-332
9. Back Matter
Pages 333-334

### 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
• 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