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We shall start with some rather dry, but necessary, technical definitions. As mentioned in the penultimate paragraph of the introduction to Part I, once you have read them you have a number of choices as to what to read next.

Basic statistics has its Chebyshev inequality, martingale theory has its maximal inequalities, Markov processes have large deviations, but all pale in comparison to the power and simplicity of the coresponding basic inequality of Gaussian processes. This inequality was discovered independently, and established with very different proofs, by Borell [30] and Tsirelson, Ibragimov, and Sudakov (TIS) [160]. For brevity, we shall call it the Borell–TIS inequality. In the following section we shall treat it in some detail.

While most of what we shall have to say in this brief chapter is rather theoretical, it actually covers one of the most important practical aspects of Gaussian modeling. The basic result is Theorem 3.1.1, which states that centered Gaussian process with a continuous covariance function has an expansion of the form where the ξ are i.i.d. (0, 1), and the ϕn are certain functions on determined by the covariance function of . In general, the convergence in (3.0.1) is in ({ℙ) for each ∈ , but (Theorem 3.1.2) if is a.s. continuous then the convergence is uniform over , with probability one.

As we have already mentioned more than once before, one of the oldest and most important problems in the study of stochastic processes of any kind is to evaluate the where is a random process over some parameter set . As usual, we shall restrict ourselves to the case in which is centered and Gaussian and is compact for the canonical metric of (1.3.1).

Stationarity has always been the backbone of almost all examples in the theory of Gaussian processes for which specific computations were possible. As described in the preface, one of the main reasons we shall be studying Gaussian processes on manifolds is to get around this assumption. Nevertheless, despite the fact that we shall ultimately try to avoid it, we invest a chapter on the topic for two reasons:

Our aim in this chapter is to develop a framework for handling , which we now redefine in a nonstochastic framework.

As we have said more than once, this chapter is intended to serve as a rapid and noncomprehensive introduction to differential geometry, basically in the format of a “glossary of terms. ” Most will be familiar to those who have taken a couple courses in differential geometry, and hopefully informative enough to allow the uninitiated to follow the calculations in later chapters. However, to go beyond merely following the arguments there and to reach the level of a real understanding of what is going on, it will be necessary to learn the material from its classical sources.

So far, all that we have had to say about manifolds and calculus on manifolds has been of a local nature; i.e., it depended only on what was happening in individual charts. However, looking back at what we did in Sections 6.1–6.3 in the setting of integral geometry, we see that this is not going to solve our main problem, which is understanding the global structure of excursion sets of random fields now defined over manifolds.

In the preceding chapter we set up the two main geometric tools that we shall need in Part III of the book. The first of these are piecewise smooth manifolds of one kind or another, which will serve there as parameter spaces for our random fields, as well as appearing in the proofs. The second are the Lipschitz–Killing curvatures that we met briefly in Chapter 7 and shall look at far more closely, in the piecewise smooth scenario, in Chapter 10. These will appear in the answers to the questions we shall ask. Between the questions and the answers will lie considerable computation, and the main geometric tool that we shall need there is the topic of this short chapter.

In Chapters 7 and 8we invested a good deal of time and energy in developing the many results we need from differential geometry. The time has now come to begin to reap the benefits of our investment, while at the same time developing some themes a little further for later exploitation. This chapter focuses on the celebrated volume-of-tubes formula of Wey1 [73, 168], which expresses the Lebesgue volume of a tube of radius ρ around a set embedded in ℝ or S(ℝ{sl}) in terms of the radius of the tube1 and the Lipschitz–Killing curvatures of (see Theorem 10.5.6). It is an interesting fact, particularly in view of the fact that this is a book about probability that is claimed to have applications to statistics, and despite the fact thatWeyl’s formula is today the basis of a large literature in geometry, that the origins of the volume-of-tubes formulas were inspired by a statistical problem. This problem, along with its solution due to Hotelling [79], were related to regression analysis and involved the one-dimensional volume-of-tubes problem on a sphere, not unrelated to the computation we shall do in a moment.