Random and Quasi-Random Point Sets

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.

These topics are of theoretical and applied interest in mathematics and statistics. There are applications to mathematical finance, cryptology, and applied statistics. This research-level monograph surveys the theoretical and applied aspects.

From Probabilistic Diophantine Approximation to Quadratic Fields.- 1 Part I: Super Irregularity.- 2 Part II: Probabilistic Diophantine Approximation.- 3 Part III: Quadratic Fields and Continued Fractions.- 4 Part IV: Class Number One Problems.- 5 Part V: Cesaro Mean of % MathType!MTEF!2!1!+-% feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqaaaaaaaaaWdbe% aadaaeqaWdaeaapeWaaeWaa8aabaWdbmaacmaapaqaa8qacaWGUbGa% eqySdegacaGL7bGaayzFaaGaeyOeI0IaaGymaiaac+cacaaIYaaaca% GLOaGaayzkaaaal8aabaWdbiaad6gaaeqaniabggHiLdaaaa!42C9!$$ sumnolimits_n {left( {left{ {nalpha } right} - 1/2} right)} $$.- 6 References.- On the Assessment of Random and Quasi-Random Point Sets.- 1 Introduction.- 2 Chapter for the Practitioner.- 3 Mathematical Preliminaries.- 4 Uniform Distribution Modulo One.- 5 The Spectral Test.- 6 The Weighted Spectral Test.- 7 Discrepancy.- 8 Summary.- 9 Acknowledgements.- 10 References.- Lattice Rules: How Well Do They Measure Up?.- 1 Introduction.- 2 Some Basic Properties of Lattice Rules.- 3 A General Approach to Worst-Case and Average-Case Error Analysis.- 4 Examples of Other Discrepancies.- 5 Shift-Invariant Kernels and Discrepancies.- 6 Discrepancy Bounds.- 7 Discrepancies of Integration Lattices and Nets.- 8 Tractability of High Dimensional Quadrature.- 9 Discussion and Conclusion.- 10 References.- Digital Point Sets: Analysis and Application.- 1 Introduction.- 2 The Concept and Basic Properties of Digital Point Sets.- 3 Discrepancy Bounds for Digital Point Sets.- 4 Special Classes of Digital Point Sets and Quality Bounds.- 5 Digital Sequences Based on Formal Laurent Series and Non-Archimedean Diophantine Approximation.- 6 Analysis of Pseudo-Random-Number Generators by Digital Nets.- 7 The Digital Lattice Rule.- 8 Outlook and Open Research Topics.- 9 References.- Random Number Generators: Selection Criteria and Testing.- 1 Introduction.- 2 Design Principles and Figures of Merit.- 3 Empirical Statistical Tests.- 4 Examples of Empirical Tests.- 5 Collections of Small RNGs.- 6 Systematic Testing for Small RNGs.- 7 How Do Real-Life Generators Fare in These Tests?.- 8 Acknowledgements.- 9 References.- Nets, (ts)-Sequences, and Algebraic Geometry.- 1 Introduction.- 2 Basic Concepts.- 3 The Digital Method.- 4 Background on Algebraic Curves over Finite Fields.- 5 Construction of (ts)-Sequences.- 6 New Constructions of (tms)-Nets.- 7 New Algebraic Curves with Many Rational Points.- 8 References.- Financial Applications of Monte Carlo and Quasi-Monte Carlo Methods.- 1 Introduction.- 2 Monte Carlo Methods for Finance Applications.- 3 Speeding Up by Quasi-Monte Carlo Methods.- 4 Future Topics.- 5 References.