Overview
- Recasts old topics in random fields by following a completely new way of handling both geometry and probability
- Significant exposition of the work of others in the field
- Excellent reference work as well as excellent work for self study
Part of the book series: Springer Monographs in Mathematics (SMM)
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About this book
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.
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Table of contents (15 chapters)
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Front Matter
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Gaussian Processes
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Front Matter
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Geometry
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Front Matter
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The Geometry of Random Fields
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Front Matter
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Back Matter
Reviews
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)
Authors and Affiliations
Bibliographic Information
Book Title: Random Fields and Geometry
Authors: Robert J. Adler, Jonathan E. Taylor
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-0-387-48116-6
Publisher: Springer New York, NY
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer-Verlag New York 2007
Hardcover ISBN: 978-0-387-48112-8Published: 12 June 2007
Softcover ISBN: 978-1-4419-2369-1Published: 25 November 2010
eBook ISBN: 978-0-387-48116-6Published: 29 January 2009
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XVIII, 454
Number of Illustrations: 21 b/w illustrations
Topics: Probability Theory and Stochastic Processes, Statistics, general, Geometry, Mathematical Methods in Physics