Level Sets and Extrema of Random Processes and Fields.
Title:
Level Sets and Extrema of Random Processes and Fields.
Author:
Azais, Jean-Marc.
ISBN:
9780470434635
Personal Author:
Edition:
1st ed.
Physical Description:
1 online resource (407 pages)
Contents:
LEVEL SETS AND EXTREMA OF RANDOM PROCESSES AND FIELDS -- CONTENTS -- PREFACE -- INTRODUCTION -- 1 CLASSICAL RESULTS ON THE REGULARITY OF PATHS -- 1.1 Kolmogorov's Extension Theorem -- 1.2 Reminder on the Normal Distribution -- 1.3 0-1 Law for Gaussian Processes -- 1.4 Regularity of Paths -- Exercises -- 2 BASIC INEQUALITIES FOR GAUSSIAN PROCESSES -- 2.1 Slepian Inequalities -- 2.2 Ehrhard's Inequality -- 2.3 Gaussian Isoperimetric Inequality -- 2.4 Inequalities for the Tails of the Distribution of the Supremum -- 2.5 Dudley's Inequality -- Exercises -- 3 CROSSINGS AND RICE FORMULAS FOR ONE-DIMENSIONAL PARAMETER PROCESSES -- 3.1 Rice Formulas -- 3.2 Variants and Examples -- Exercises -- 4 SOME STATISTICAL APPLICATIONS -- 4.1 Elementary Bounds for P{M >u} -- 4.2 More Detailed Computation of the First Two Moments -- 4.3 Maximum of the Absolute Value -- 4.4 Application to Quantitative Gene Detection -- 4.5 Mixtures of Gaussian Distributions -- Exercises -- 5 THE RICE SERIES -- 5.1 The Rice Series -- 5.2 Computation of Moments -- 5.3 Numerical Aspects of the Rice Series -- 5.4 Processes with Continuous Paths -- 6 RICE FORMULAS FOR RANDOM FIELDS -- 6.1 Random Fields from R(d) to R(d) -- 6.2 Random Fields from R(d) to R(d)́, d > d ́-- Exercises -- 7 REGULARITY OF THE DISTRIBUTION OF THE MAXIMUM -- 7.1 Implicit Formula for the Density of the Maximum -- 7.2 One-Parameter Processes -- 7.3 Continuity of the Density of the Maximum of Random Fields -- Exercises -- 8 THE TAIL OF THE DISTRIBUTION OF THE MAXIMUM -- 8.1 One-Dimensional Parameter: Asymptotic Behavior of the Derivatives of F(M) -- 8.2 An Application to Unbounded Processes -- 8.3 A General Bound for p(M) -- 8.4 Computing (x) for Stationary Isotropic Gaussian Fields -- 8.5 Asymptotics as x + -- 8.6 Examples -- Exercises -- 9 THE RECORD METHOD -- 9.1 Smooth Processes with One-Dimensional Parameters.
9.2 Nonsmooth Gaussian Processes -- 9.3 Two-Parameter Gaussian Processes -- Exercises -- 10 ASYMPTOTIC METHODS FOR AN INFINITE TIME HORIZON -- 10.1 Poisson Character of High Up-Crossings -- 10.2 Central Limit Theorem for Nonlinear Functionals -- Exercises -- 11 GEOMETRIC CHARACTERISTICS OF RANDOM SEA WAVES -- 11.1 Gaussian Model for an Infinitely Deep Sea -- 11.2 Some Geometric Characteristics of Waves -- 11.3 Level Curves, Crests, and Velocities for Space Waves -- 11.4 Real Data -- 11.5 Generalizations of the Gaussian Model -- Exercises -- 12 SYSTEMS OF RANDOM EQUATIONS -- 12.1 The Shub-Smale Model -- 12.2 More General Models -- 12.3 Noncentered Systems (Smoothed Analysis) -- 12.4 Systems Having a Law Invariant Under Orthogonal Transformations and Translations -- 13 RANDOM FIELDS AND CONDITION NUMBERS OF RANDOM MATRICES -- 13.1 Condition Numbers of Non-Gaussian Matrices -- 13.2 Condition Numbers of Centered Gaussian Matrices -- 13.3 Noncentered Gaussian Matrices -- REFERENCES AND SUGGESTED READING -- NOTATION -- INDEX.
Abstract:
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.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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Electronic Access:
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