Random and Quasi-Random Point Sets

Transcription

1 Peter Hellekalek Gerhard Larcher (Editors) Random and Quasi-Random Point Sets With contributions by JözsefBeck Peter Hellekalek Fred J. Hickemell Gerhard Larcher Pierre L'Ecuyer Harald Niederreiter Shu Tezuka Chaoping Xing Springer

2 Contents Preface v From Probabilistic Diophantine Approximation to Quadratic Fields 1 Jozsef Beck 1 Part I: Super Irregularity 1 2 Part II: Probabilistic Diophantine Approximation Local Case: Inhomogeneous Pell Inequalities - Hyperbolas Beyond Quadratic Irrationals Global Case: Lattice Points in Tilted Rectangles Simultaneous Case 13 3 Part III: Quadratic Fields and Continued Fractions Cesaro Mean of ^{na 1 / 2 } and Quadratic Fields Hardy-Littlewood Lemma Part IV: Class Number One Problems An Attempt to Reduce the Yokoi's Conjecture to a Finite Amount of Computation 27 5 Part V: Cesaro Mean of ^J{na} - 1/2) 32 6 References 46 On the Assessment of Random and Quasi-Random Point Sets 49 Peter Hellekalek 1 Introduction 49 2 Chapter for the Practitioner Assessing RNGs Correlation Analysis for RNGs I Correlation Analysis for RNGs II Theory vs. Practice I: Leap-Frog Streams Theory vs. Practice II: Parallel Monte Carlo Integration Assessing LDPs Good Lattice Points GLPs vs. (t,m,s)-nets Conclusion 69 3 Mathematical Preliminaries Haar and Walsh Series Integration Lattices 74 4 Uniform Distribution Modulo One The Definition of Uniformly Distributed Sequences 76

3 x Contents 4.2 Weyl Sums and Weyl's Criterion Remarks 78 5 The Spectral Test Definition Properties Examples Geometrie Interpretation Remarks 88 6 The Weighted Spectral Test Definition Examples and Properties Remarks 93 7 Discrepancy Definition The Inequality of Erdös-Turän-Koksma Remarks 98 8 Summary 99 9 Acknowledgements References 101 Lattice Rules: How Well Do They Measure Up? 109 Fred J. Hickernell 1 Introduction Some Basic Properties of Lattice Rules A General Approach to Worst-Case and Average-Case Error Analysis Worst-Case Quadrature Error for Reproducing Kernel Hubert Spaces A More General Worst-Case Quadrature Error Analysis Average-Case Quadrature Error Analysis Examples of Other Discrepancies The ANOVA Decomposition A Generalization of P a (L) with Weights The Periodic Bernoulli Discrepancy - Another Generalization of P a (L) The Non-Periodic Bernoulli Discrepancy The Star Discrepancy The Unanchored Discrepancy The Wrap-Around Discrepancy The Symmetrie Discrepancy Shift-Invariant Kernels and Discrepancies Discrepancy Bounds Upper Bounds for P a (L) A Lower Bound on üjr, a,i(p) 145

4 Contents xi 6.3 Quadrature Rules with Different Weights Copy Rules Discrepancies of Integration Lattices and Nets The Expected Discrepancy of Randomized (0, m, s)-nets Infinite Sequences of Embedded Lattices Tractability of High Dimensional Quadrature Quadrature in Arbitrarily High Dimensions The Effective Dimension of an Integrand Discussion and Conclusion References 163 Digital Point Sets: Analysis and Application 167 Gerhard Larcher 1 Introduction The Concept and Basic Properties of Digital Point Sets Discrepancy Bounds for Digital Point Sets Special Classes of Digital Point Sets and Quality Bounds Digital Sequences Based on Formal Laurent Series and Non-Archimedean Diophantine Approximation Analysis of Pseudo-Random-Number Generators by Digital Nets The Digital Lattice Rule Outlook and Open Research Topics References 217 Random Number Generators: Selection Criteria and Testing 223 Pierre L 'Ecuyer and Peter Hellekalek 1 Introduction Design Principles and Figures of Merit A Roulette Wheel Sampling from $ ( The Lattice Structure of MRG's Equidistribution for Regulär Partitions in Cubic Boxes Other Measures of Divergence Empirical Statistical Tests What are the Good Tests? Two-Level Tests Collections of Empirical Tests Examples of Empirical Tests Serial Tests of Equidistribution Tests Based on Close Points in Space Collections of Small RNGs Small Linear Congruential Generators 237

5 xii Contents 5.2 Explicit Inversive Congruential Generators Compound Cubic Congruential Generators Systematic Testing for Small RNGs Serial Tests of Equidistribution for LCGs Serial Tests of Equidistribution for Nonlinear Generators A Summary of the Serial Tests Results Close-Pairs Tests for LCGs Close-Pairs Tests for Nonlinear Generators A Summary of the Close-Pairs Tests Results How Do Real-Life Generators Fare in These Tests? Acknowledgements References 259 Nets, (t, s)-sequences, and Algebraic Geometry 267 Harald Niederreiter and Chaoping Xing 1 Introduction Basic Concepts The Digital Method Background on Algebraic Curves over Finite Fields Construction of (t, s)-sequences New Constructions of (t, m, s)-nets New Algebraic Curves with Many Rational Points References 299 Financial Applications of Monte Carlo and Quasi-Monte Carlo Methods 303 Shu Tezuka 1 Introduction Monte Carlo Methods for Finance Applications Preliminaries for Derivative Pricing Variance Reduction Techniques Caveats for Computer Implementation Speeding Up by Quasi-Monte Carlo Methods What are Quasi-Monte Carlo Methods? Generalized Faure Sequences Numerical Experiments Discussions Future Topics Monte Carlo Simulations for American Options Research Issues Related to Quasi-Monte Carlo Methods References 329