Random and QuasiRandom Point Sets (Lecture Notes in Statistics, 138)

This volume is a collection of survey papers on recent developments in the fields of quasiMonte 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, socalled 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 highdimensional random point sets. The degree of uniformity is strongly related to the degree of correlations within the random numbers. The quasiMonte Carlo approach aims at improving the rate of conver gence in the Monte Carlo method by numbertheoretic techniques. It yields deterministic bounds for the approximation error. The main mathematical tool here are socalled lowdiscrepancy sequences. These quasirandom points are produced by deterministic algorithms and should be as super uniformly distributed as possible. Hence, both in uniform random number generation and in quasiMonte Carlo methods, we study the uniformity of deterministically generated point sets in high dimensions. By a (common) abuse oflanguage, one speaks of random and quasirandom point sets. The central questions treated in this book are (i) how to generate, (ii) how to analyze, and (iii) how to apply such highdimensional point sets.