Random Fields and Geometry (eBook)

R. J. Adler, Jonathan E. Taylor (Autoren)

eBook Download: PDF

2009 | 2007
XVIII, 454 Seiten
Springer New York (Verlag)
978-0-387-48116-6 (ISBN)

Infos

This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined.

'Random Fields and Geometry' will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory.

 

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Since the term "e;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.

Inhalt

Preface 6
Contents 13
Part I Gaussian Processes 18
1 Gaussian Fields 22
2 Gaussian Inequalities 64
3 Orthogonal Expansions 80
4 Excursion Probabilities 90
5 Stationary Fields 115
Part II Geometry 136
6 Integral Geometry 139
7 Differential Geometry 160
8 Piecewise Smooth Manifolds 193
9 Critical Point Theory 202
10 Volume of Tubes 222
Part III The Geometry of Random Fields 267
11 Random Fields on Euclidean Spaces 270
12 Random Fields on Manifolds 307
13 Mean Intrinsic Volumes 337
14 Excursion Probabilities for Smooth Fields 355
15 Non-Gaussian Geometry 393
References 440
Notation Index 448
Subject Index 450

Produkt-Details

• Erscheint lt. Verlag: 29.1.2009
• Reihe/Serie: Springer Monographs in Mathematics
• Reihe/Serie: Springer Monographs in Mathematics
• Zusatzinfo: XVIII, 454 p. 21 illus.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: area • astrophysics • Differential Geometry • Gaussian process • Geometry • Probability • Volume
• ISBN-10: 0-387-48116-8 / 0387481168
• ISBN-13: 978-0-387-48116-6 / 9780387481166