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Random Fields and Geometry

BuchGebunden
454 Seiten
Englisch
Springererschienen am12.06.2007
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.mehr
Verfügbare Formate
BuchGebunden
EUR149,79
BuchKartoniert, Paperback
EUR149,79
E-BookPDF1 - PDF WatermarkE-Book
EUR139,09

Produkt

KlappentextSince 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.
Zusammenfassung
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
Details
ISBN/GTIN978-0-387-48112-8
ProduktartBuch
EinbandartGebunden
Verlag
Erscheinungsjahr2007
Erscheinungsdatum12.06.2007
ReiheSpringer Monographs in Mathematics
Seiten454 Seiten
SpracheEnglisch
Gewicht878 g
IllustrationenXVIII, 454 p. 21 illus.
Artikel-Nr.10740657

Inhalt/Kritik

Inhaltsverzeichnis
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.mehr
Kritik
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 bookshelf of all probabilists interested in Gaussian processes, random fields and their statistical applications." (Ilya S. Molchanov, Zentralblatt MATH, Vol. 1149, 2008)mehr