Level Sets and Extrema of Random Processes and Fields

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.