Level sets and extrema of random processes and fields

Level Sets and Extrema of Random Processes and Fields discusses how to understand the properties of the level sets of paths as well as how to compute the probability distribution of its extremal values, which are two general classes of problems that arise in the study of random processes and fields and in related applications. This book provides a unified and accessible approach to these two topics and their relationship to classical theory and Gaussian processes and fields, and the most modern research findings are also discussed.

The authors begin with an introduction to the basic concepts of stochastic processes, including a modern review of Gaussian fields and their classical inequalities. Subsequent chapters are devoted to Rice formulas, regularity properties, and recent results on the tails of the distribution of the maximum. Finally, applications of random fields to various areas of mathematics are provided, specifically to systems of random equations and condition numbers of random matrices.

Throughout the book, applications are illustrated from various areas of study such as statistics, genomics, and oceanography while other results are relevant to econometrics, engineering, and mathematical physics. The presented material is reinforced by end-of-chapter exercises that range in varying degrees of difficulty. Most fundamental topics are addressed in the book, and an extensive, up-to-date bibliography directs readers to existing literature for further study.

Level Sets and Extrema of Random Processes and Fields is an excellent book for courses on probability theory, spatial statistics, Gaussian fields, and probabilistic methods in real computation at the upper-undergraduate and graduate levels. It is also a valuable reference for professionals in mathematics and applied fields such as statistics, engineering, econometrics, mathematical physics, and biology.

Author Notes

Jean-Marc Azais, PhD, is Professor in the Institute of Mathematics at the Universit de Toulouse, France. Dr. Azaiumlet;s has authored numerous journal articles in his areas of research interest, which include probability theory, statistical modeling, biometrics, and the design of experiments.
Mario Wschebor, PhD, is Professor in the Center of Mathematics at the Universidad de la Repblica, Uruguay. In addition to serving as President of the International Center for Pure and Applied Mathematics, Dr. Wschebor is the coauthor of numerous journal articles in the areas of random fields, stochastic analysis, random matrices, and algorithm complexity.

Preface	p. ix
Introduction	p. 1
1 Classical Results On The Regularity Of Paths	p. 10
1.1 Kolmogorov's Extension Theorem	p. 11
1.2 Reminder on the Normal Distribution	p. 14
1.3 0-1 Law for Gaussian Processes	p. 18
1.4 Regularity of Paths	p. 20
Exercises	p. 38
2 Basic Inequalities For Gaussian Processes	p. 43
2.1 Slepian Inequalities	p. 44
2.2 Ehrhard's Inequality	p. 48
2.3 Gaussian Isoperimetric Inequality	p. 52
2.4 Inequalities for the Tails of the Distribution of the Supremum	p. 53
2.5 Dudley's Inequality	p. 62
Exercises	p. 66
3 Crossings And Rice Formulas For One-Dimensional Parameter Processes	p. 68
3.1 Rice Formulas	p. 68
3.2 Variants and Examples	p. 79
Exercises	p. 86
4 Some Statistical Applications	p. 92
4.1 Elementary Bounds for P{{M > u}}	p. 93
4.2 More Detailed Computation of the First Two Moments	p. 99
4.3 Maximum of the Absolute Value	p. 105
4.4 Application to Quantitative Gene Detection	p. 106
4.5 Mixtures of Gaussian Distributions	p. 120
Exercises	p. 130
5 The Rice Series	p. 133
5.1 The Rice Series	p. 134
5.2 Computation of Moments	p. 141
5.3 Numerical Aspects of the Rice Series	p. 149
5.4 Processes with Continuous Paths	p. 155
6 Rice Formulas for Random Fields	p. 160
6.1 Random Fields from Rd to Rd	p. 161
6.2 Random Fields from Rd to Rd$$, d > d$$	p. 177
Exercises	p. 181
7 Regularity Of The Distribution of The Maximum	p. 184
7.1 Implicit Formula for the Density of the Maximum	p. 185
7.2 One-Parameter Processes	p. 188
7.3 Continuity of the Density of the Maximum of Random Fields	p. 201
Exercises	p. 203
8 The Tail Of The Distribution Of The Maximum	p. 206
8.1 One-Dimensional Parameter: Asymptotic Behavior of the Derivatives of FM	p. 208
8.2 An Application to Unbounded Processes	p. 212
8.3 A General Bound for PM	p. 216
8.4 Computing &pbar;(x) for Stationary Isotropic Gaussian Fields	p. 218
8.5 Asymptotics as x $$	p. 225
8.6 Examples	p. 237
Exercises	p. 243
9 The Record Method	p. 245
9.1 Smooth Processes with One-Dimensional Parameters	p. 245
9.2 Nonsmooth Gaussian Processes	p. 250
9.3 Two-Parameter Gaussian Processes	p. 252
Exercises	p. 256
10 Asymptotic Methods For An Infinite Time Horizon	p. 258
10.1 Poisson Character of High Up-Crossings	p. 258
10.2 Central Limit Theorem for Nonlinear Functionals	p. 267
Exercises	p. 284
11 Geometric Characteristics of Random Sea Waves	p. 285
11.1 Gaussian Model for an Infinitely Deep Sea	p. 285
11.2 Some Geometric Characteristics of Waves	p. 287
11.3 Level Curves, Crests, and Velocities for Space Waves	p. 289
11.4 Real Data	p. 297
11.5 Generalizations of the Gaussian Model	p. 298
Exercises	p. 300
12 Systems of Random Equations	p. 301
12.1 The Shub-Smale Model	p. 303
12.2 More General Models	p. 313
12.3 Noncentered Systems (Smoothed Analysis)	p. 321
12.4 Systems Having a Law Invariant Under Orthogonal Transformations and Translations	p. 333
13 Random Fields And Condition Numbers Of Random Matrices	p. 340
13.1 Condition Numbers of Non-Gaussian Matrices	p. 343
13.2 Condition Numbers of Centered Gaussian Matrices	p. 349
13.3 Noncentered Gaussian Matrices	p. 364
References And Suggested Reading	p. 373
Notation	p. 389
Index	p. 391

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