Level Sets and Extrema of Random Processes and Fields (eBook)

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.

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.

"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)