Random Fields and Geometry
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R. J. Adler, Jonathan E. Taylor
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Random Fields and Geometry
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Springer-Verlag
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9780387481166
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1
:
CHF 123.40
:
:
Wahrscheinlichkeitstheorie, Stochastik, Mathematische Statistik
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English
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454
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Wasserzeichen
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PC/MAC/eReader/Tablet
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PDF
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.
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
