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Level Sets and Extrema of Random Processes and Fields

ISBN: 978-0-470-40933-6
393 pages
March 2009
Level Sets and Extrema of Random Processes and Fields (0470409339) cover image
A timely and comprehensive treatment of random field theory with applications across diverse areas of study

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.

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Introduction.

Reading diagram.

Chapter 1: Classical results on the regularity of the paths.

1. Kolmogorov’s Extension Theorem.

2. Reminder on the Normal Distribution.

3. 0-1 law for Gaussian processes.

4. Regularity of the paths.

Exercises.

Chapter 2: Basic Inequalities for Gaussian Processes.

1. Slepian type inequalities.

2. Ehrhard’s inequality.

3. Gaussian isoperimetric inequality.

4. Inequalities for the tails of the distribution of the supremum.

5. Dudley’s inequality.

Exercises.

Chapter 3: Crossings and Rice formulas for 1-dimensional parameter processes.

1. Rice Formulas.

2. Variants and Examples.

Exercises.

Chapter 4: Some Statistical Applications.

1. Elementary bounds for P{M > u}.

2. More detailed computation of the first two moments.

3. Maximum of the absolute value.

4. Application to quantitative gene detection.

5. Mixtures of Gaussian distributions.

Exercises.

Chapter 5: The Rice Series.

1. The Rice Series.

2. Computation of Moments.

3. Numerical aspects of Rice Series.

4. Processes with Continuous Paths.

Chapter 6: Rice formulas for random fields.

1. Random fields from Rd to Rd.

2. Random fields from Rd to Rd!, d> d!.

Exercises.

Chapter 7: Regularity of the Distribution of the Maximum.

1. The implicit formula for the density of the maximum.

2. One parameter processes.

3. Continuity of the density of the maximum of random fields.

Exercises.

Chapter 8: The tail of the distribution of the maximum.

1. One-dimensional parameter: asymptotic behavior of the derivatives of FM.

2. An Application to Unbounded Processes.

3. A general bound for pM.

4. Computing p(x) for stationary isotropic Gaussian fields.

5. Asymptotics as x! +".

6. Examples.

Exercises.

Chapter 9: The record method.

1. Smooth processes with one dimensional parameter.

2. Non-smooth Gaussian processes.

3. Two-parameter Gaussian processes.

Exercises.

Chapter 10: Asymptotic methods for infinite time horizon.

1. Poisson character of "high" up-crossings.

2. Central limit theorem for non-linear functionals.

Exercises.

Chapter 11: Geometric characteristics of random sea-waves.

1. Gaussian model for infinitely deep sea.

2. Some geometric characteristics of waves.

3. Level curves, crests and velocities for space waves.

4. Real Data.

5. Generalizations of the Gaussian model.

Exercises.

Chapter 12: Systems of random equations.

1. The Shub-Smale model.

2. More general models.

3. Non-centered systems (smoothed analysis).

4. Systems having a law invariant under orthogonal transformations and translations.

Chapter 13: Random fields and condition numbers of random matrices.

1. Condition numbers of non-Gaussian matrices.

2. Condition numbers of centered Gaussian matrices.

3. Non-centered Gaussian matrices.

Notations.

References.

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Jean-Marc Aza┐s, PhD, is Professor in the Institute of Mathematics at the Université de Toulouse, France. Dr. Azaïs 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 República, 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.

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"It is a very original book, distinguished by its topic and its ability to make use of intuitive basic techniques, such as the Rice formula for instance. So we can say already that it is one of the most important books in probability theory published in the last twenty years." (Zentralblatt Math, 2010)

 

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