Random Fields and Geometry

Since the term "random ?eld'' has a variety of different connotations, ranging from agriculture to statistical mechanics, let us start by clarifying that, in this book, a random ?eld is a stochastic process, usually taking values in a Euclidean space, and de?ned over a parameter space of dimensionality at least 1. Consequently, random processes de?ned on countable parameter spaces will not 1 appear here. Indeed, even processes on R will make only rare appearances and, from the point of view of this book, are almost trivial. The parameter spaces we like best are manifolds, although for much of the time we shall require no more than that they be pseudometric spaces. With this clari?cation in hand, the next thing that you should know is that this book will have a sequel dealing primarily with applications. In fact, as we complete this book, we have already started, together with KW (Keith Worsley), on a companion volume [8] tentatively entitled RFG-A,or Random Fields and Geometry: Applications. The current volume-RFG-concentrates on the theory and mathematical background of random ?elds, while RFG-A is intended to do precisely what its title promises. Once the companion volume is published, you will ?nd there not only applications of the theory of this book, but of (smooth) random ?elds in general.

List of contents

Gaussian Processes.- Gaussian Fields.- Gaussian Inequalities.- Orthogonal Expansions.- Excursion Probabilities.- Stationary Fields.- Geometry.- Integral Geometry.- Differential Geometry.- Piecewise Smooth Manifolds.- Critical Point Theory.- Volume of Tubes.- The Geometry of Random Fields.- Random Fields on Euclidean Spaces.- Random Fields on Manifolds.- Mean Intrinsic Volumes.- Excursion Probabilities for Smooth Fields.- Non-Gaussian Geometry.

Summary

Since the term “random ?eld’’ has a variety of different connotations, ranging from agriculture to statistical mechanics, let us start by clarifying that, in this book, a random ?eld is a stochastic process, usually taking values in a Euclidean space, and de?ned over a parameter space of dimensionality at least 1. Consequently, random processes de?ned on countable parameter spaces will not 1 appear here. Indeed, even processes on R will make only rare appearances and, from the point of view of this book, are almost trivial. The parameter spaces we like best are manifolds, although for much of the time we shall require no more than that they be pseudometric spaces. With this clari?cation in hand, the next thing that you should know is that this book will have a sequel dealing primarily with applications. In fact, as we complete this book, we have already started, together with KW (Keith Worsley), on a companion volume [8] tentatively entitled RFG-A,or Random Fields and Geometry: Applications. The current volume—RFG—concentrates on the theory and mathematical background of random ?elds, while RFG-A is intended to do precisely what its title promises. Once the companion volume is published, you will ?nd there not only applications of the theory of this book, but of (smooth) random ?elds in general.

Additional text

From the reviews:

Developing good bounds for the distribution of the suprema of a Gaussian field $f$, i.e., for the quantity $\Bbb{P}\{\sup_{t\in M}f(t)\ge u}$, has been for a long time both a difficult and an interesting subject of research. A thorough presentation of this problem is the main goal of the book under review, as is stated by the authors in its preface. The authors develop their results in the context of smooth Gaussian fields, where the parameter spaces $M$ are Riemannian stratified manifolds, and their approach is of a geometrical nature. The book is divided into three parts. Part I is devoted to the presentation of the necessary tools of Gaussian processes and fields. Part II concisely exposes the required prerequisites of integral and differential geometry. Finally, in part III, the kernel of the book, a formula for the expectation of the Euler characteristic function of an excursion set and its approximation to the distribution of the maxima of the field, is precisely established. The book is written in an informal style, which affords a very pleasant reading. Each chapter begins with a presentation of the matters to be addressed, and the footnotes, located throughout the text, serve as an indispensable complement and many times as historical references. The authors insist on the fact that this book should not only be considered as a theoretical adventure and they recommend a second volume where they develop indispensable applications which highlight all the power of their results. (José Rafael León for Mathematical Reviews)

"This book presents the modern theory of excursion probabilities and the geometry of excursion sets for … random fields defined on manifolds. ... The book is understandable for students … with a good background in analysis. ... The interdisciplinary nature of this book, the beauty and depth of the presented mathematical theory make it an indispensable part of every mathematical library and a bookshelfof all probabilists interested in Gaussian processes, random fields and their statistical applications." (Ilya S. Molchanov, Zentralblatt MATH, Vol. 1149, 2008)

Report

From the reviews:
Developing good bounds for the distribution of the suprema of a Gaussian field $f$, i.e., for the quantity $Bbb{P}{sup_{tin M}f(t)ge u}$, has been for a long time both a difficult and an interesting subject of research. A thorough presentation of this problem is the main goal of the book under review, as is stated by the authors in its preface. The authors develop their results in the context of smooth Gaussian fields, where the parameter spaces $M$ are Riemannian stratified manifolds, and their approach is of a geometrical nature. The book is divided into three parts. Part I is devoted to the presentation of the necessary tools of Gaussian processes and fields. Part II concisely exposes the required prerequisites of integral and differential geometry. Finally, in part III, the kernel of the book, a formula for the expectation of the Euler characteristic function of an excursion set and its approximation to the distribution of the maxima of the field, is precisely established. The book is written in an informal style, which affords a very pleasant reading. Each chapter begins with a presentation of the matters to be addressed, and the footnotes, located throughout the text, serve as an indispensable complement and many times as historical references. The authors insist on the fact that this book should not only be considered as a theoretical adventure and they recommend a second volume where they develop indispensable applications which highlight all the power of their results. (José Rafael León for Mathematical Reviews)
"This book presents the modern theory of excursion probabilities and the geometry of excursion sets for ... random fields defined on manifolds. ... The book is understandable for students ... with a good background in analysis. ... The interdisciplinary nature of this book, the beauty and depth of the presented mathematical theory make it an indispensable part of every mathematical library and a bookshelfof all probabilists interested in Gaussian processes, random fields and their statistical applications." (Ilya S. Molchanov, Zentralblatt MATH, Vol. 1149, 2008)

Product details

• Authors: R Adler, R J Adler, R. J. Adler, Robert J. Adler, Jonathan Taylor, Jonathan E Taylor, Jonathan E. Taylor
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 27.10.2010
• EAN: 9781441923691
• ISBN: 978-1-4419-2369-1
• No. of pages: 454
• Dimensions: 157 mm x 26 mm x 236 mm
• Weight: 709 g
• Illustrations: XVIII, 454 p. 21 illus.
• Series: Springer Monographs in Mathematics; Springer Monographs in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics; A, Statistics, geometry, Physics, Mathematics and Statistics, Probability Theory and Stochastic Processes, Mathematical physics, Probability & statistics, Probabilities, Stochastics, Probability Theory, Mathematical Methods in Physics, Statistics, general