Summary: Part I
If you have not yet read the preface, then please do so now.
Since you have read the preface, you already know a number of important things
about this book, including the fact that Part I is about Gaussian random fields.
The centrality of Gaussian fields to this book is due to two basic factors:
· Gaussian processes have a rich, detailed, and very well-understood general theory,
which makes them beloved by theoreticians.
· In applications of random field theory, as in applications of almost any theory, it is
important to have specific, explicit formulas that allow one to predict, to compare
theory with experiment, etc. As we shall see in Part III, it will be only for Gaussian
(and related; cf. Section 1.4.6 and Chapter 15) fields that it is possible to derive
such formulas, and then only in the setting of excursion sets.
The main reason behind both these facts is the convenient analytic form of the
multivariate Gaussian density, and the related properties of Gaussian fields. This is
what Part I is about.
There are five main collections of basic results that will be of interest to us.
Rather interestingly, although later in the book we shall be interested in Gaussian
fields defined over various types of manifolds, the basic theory of Gaussian fields