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
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 4 2.1 Local Case: Inhomogeneous Pell Inequalities - Hyperbolas 5 2.2 Beyond Quadratic Irrationals 9 2.3 Global Case: Lattice Points in Tilted Rectangles 9 2.4 Simultaneous Case 13 3 Part III: Quadratic Fields and Continued Fractions 13 3.1 Cesaro Mean of ^{na 1 / 2 } and Quadratic Fields 17 3.2 Hardy-Littlewood Lemma 14 19 4 Part IV: Class Number One Problems 22 4.1 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 50 2.1 Assessing RNGs 51 2.2 Correlation Analysis for RNGs I 53 2.3 Correlation Analysis for RNGs II 56 2.4 Theory vs. Practice I: Leap-Frog Streams 58 2.5 Theory vs. Practice II: Parallel Monte Carlo Integration 61 2.6 Assessing LDPs 65 2.7 Good Lattice Points 67 2.8 GLPs vs. (t,m,s)-nets 68 2.9 Conclusion 69 3 Mathematical Preliminaries 70 3.1 Haar and Walsh Series 70 3.2 Integration Lattices 74 4 Uniform Distribution Modulo One 76 4.1 The Definition of Uniformly Distributed Sequences 76
x Contents 4.2 Weyl Sums and Weyl's Criterion 77 4.3 Remarks 78 5 The Spectral Test 79 5.1 Definition 81 5.2 Properties 82 5.3 Examples 83 5.4 Geometrie Interpretation 84 5.5 Remarks 88 6 The Weighted Spectral Test 90 6.1 Definition 90 6.2 Examples and Properties 91 6.3 Remarks 93 7 Discrepancy 94 7.1 Definition 94 7.2 The Inequality of Erdös-Turän-Koksma 95 7.3 Remarks 98 8 Summary 99 9 Acknowledgements 100 10 References 101 Lattice Rules: How Well Do They Measure Up? 109 Fred J. Hickernell 1 Introduction 109 2 Some Basic Properties of Lattice Rules 110 3 A General Approach to Worst-Case and Average-Case Error Analysis 115 3.1 Worst-Case Quadrature Error for Reproducing Kernel Hubert Spaces 116 3.2 A More General Worst-Case Quadrature Error Analysis 120 3.3 Average-Case Quadrature Error Analysis 123 4 Examples of Other Discrepancies 125 4.1 The ANOVA Decomposition 125 4.2 A Generalization of P a (L) with Weights 128 4.3 The Periodic Bernoulli Discrepancy - Another Generalization of P a (L) 131 4.4 The Non-Periodic Bernoulli Discrepancy 132 4.5 The Star Discrepancy 133 4.6 The Unanchored Discrepancy 135 4.7 The Wrap-Around Discrepancy 136 4.8 The Symmetrie Discrepancy 138 5 Shift-Invariant Kernels and Discrepancies 139 6 Discrepancy Bounds 142 6.1 Upper Bounds for P a (L) 143 6.2 A Lower Bound on üjr, a,i(p) 145
Contents xi 6.3 Quadrature Rules with Different Weights 148 6.4 Copy Rules 149 7 Discrepancies of Integration Lattices and Nets 152 7.1 The Expected Discrepancy of Randomized (0, m, s)-nets 152 7.2 Infinite Sequences of Embedded Lattices 153 8 Tractability of High Dimensional Quadrature 154 8.1 Quadrature in Arbitrarily High Dimensions 154 8.2 The Effective Dimension of an Integrand 155 9 Discussion and Conclusion 158 10 References 163 Digital Point Sets: Analysis and Application 167 Gerhard Larcher 1 Introduction 167 2 The Concept and Basic Properties of Digital Point Sets 169 3 Discrepancy Bounds for Digital Point Sets 177 4 Special Classes of Digital Point Sets and Quality Bounds 183 5 Digital Sequences Based on Formal Laurent Series and Non-Archimedean Diophantine Approximation 188 6 Analysis of Pseudo-Random-Number Generators by Digital Nets.. 194 7 The Digital Lattice Rule 197 8 Outlook and Open Research Topics 210 9 References 217 Random Number Generators: Selection Criteria and Testing 223 Pierre L 'Ecuyer and Peter Hellekalek 1 Introduction 223 2 Design Principles and Figures of Merit 224 2.1 A Roulette Wheel 224 2.2 Sampling from $ ( 225 2.3 The Lattice Structure of MRG's 226 2.4 Equidistribution for Regulär Partitions in Cubic Boxes 228 2.5 Other Measures of Divergence 230 3 Empirical Statistical Tests 231 3.1 What are the Good Tests? 231 3.2 Two-Level Tests 232 3.3 Collections of Empirical Tests 232 4 Examples of Empirical Tests 233 4.1 Serial Tests of Equidistribution 233 4.2 Tests Based on Close Points in Space 236 5 Collections of Small RNGs 237 5.1 Small Linear Congruential Generators 237
xii Contents 5.2 Explicit Inversive Congruential Generators 237 5.3 Compound Cubic Congruential Generators 238 6 Systematic Testing for Small RNGs 239 6.1 Serial Tests of Equidistribution for LCGs 239 6.2 Serial Tests of Equidistribution for Nonlinear Generators 245 6.3 A Summary of the Serial Tests Results 245 6.4 Close-Pairs Tests for LCGs 254 6.5 Close-Pairs Tests for Nonlinear Generators 255 6.6 A Summary of the Close-Pairs Tests Results 255 7 How Do Real-Life Generators Fare in These Tests? 258 8 Acknowledgements 259 9 References 259 Nets, (t, s)-sequences, and Algebraic Geometry 267 Harald Niederreiter and Chaoping Xing 1 Introduction 267 2 Basic Concepts 268 3 The Digital Method 271 4 Background on Algebraic Curves over Finite Fields 274 5 Construction of (t, s)-sequences 275 6 New Constructions of (t, m, s)-nets 284 7 New Algebraic Curves with Many Rational Points 292 8 References 299 Financial Applications of Monte Carlo and Quasi-Monte Carlo Methods 303 Shu Tezuka 1 Introduction 303 2 Monte Carlo Methods for Finance Applications 305 2.1 Preliminaries for Derivative Pricing 305 2.2 Variance Reduction Techniques 306 2.3 Caveats for Computer Implementation 311 3 Speeding Up by Quasi-Monte Carlo Methods 313 3.1 What are Quasi-Monte Carlo Methods? 313 3.2 Generalized Faure Sequences 316 3.3 Numerical Experiments 320 3.4 Discussions 325 4 Future Topics 326 4.1 Monte Carlo Simulations for American Options 326 4.2 Research Issues Related to Quasi-Monte Carlo Methods 328 5 References 329