Random and Quasi-Random Point Sets

1. Front Matter

   Pages i-xii

   PDF

2. From Probabilistic Diophantine Approximation to Quadratic Fields

   • József Beck

   Pages 1-48

3. On the Assessment of Random and Quasi-Random Point Sets

   • Peter Hellekalek

   Pages 49-108

4. Lattice Rules: How Well Do They Measure Up?

   • Fred J. Hickernell

   Pages 109-166

5. Digital Point Sets: Analysis and Application

   • Gerhard Larcher

   Pages 167-222

6. Random Number Generators: Selection Criteria and Testing

   • Pierre L’Ecuyer, Peter Hellekalek

   Pages 223-265

7. Nets, (t, s)-Sequences, and Algebraic Geometry

   • Harald Niederreiter, Chaoping Xing

   Pages 267-302

8. Financial Applications of Monte Carlo and Quasi-Monte Carlo Methods

   • Shu Tezuka

   Pages 303-332

9. Back Matter

   Pages 333-334

   PDF

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.

• Generator
• Kernel
• LDA
• Probability theory
• statistics

• Institut für Mathematik, Universität Salzburg, Salzburg, Austria

  Peter Hellekalek, Gerhard Larcher

Policies and ethics