##
**Random fields and geometry.**
*(English)*
Zbl 1149.60003

Springer Monographs in Mathematics. New York, NY: Springer (ISBN 978-0-387-48112-8/hbk; 978-1-4419-2369-1/pbk; 978-0-387-48116-6/ebook). xvii, 448 p. (2007).

This book presents the modern theory of excursion probabilities and the geometry of excursion sets for rather general smooth random fields defined on manifolds. Namely, let \(f\) be a smooth, \(\mathbb{R}^k\)-valued Gaussian random field over piecewise smooth manifold \(M\). Its excursion set is defined by
\[
A_D=A_D(f,M)=\{t\in M;\; f(t)\in D\}\,,
\]
where \(D\) is a subset of \(\mathbb{R}^k\). The basic geometric properties of the excursion set are described by its Euler-Poincaré (also called Euler) characteristic \(\phi(A_D)\). The Euler-Poincaré characteristic \(\phi\) takes value zero on empty set, value one on convex sets and then extended by additivity to the convex ring. In also can be defined in terms of the homology class of a set and so applies to excursion sets. The cornerstone of the presented theory is the decomposition formula for the expected Euler-Poincaré characteristic of the excursion set
\[
\mathbb{E} \phi(A_D)=\sum_{j=0}^{\mathrm{dim}(M)} (2\pi)^{-j/2} \mathcal{L}_j(M)\mathcal{M}_j^k(D)\,,
\]
where \(\mathcal{L}_j(M)\) are the Lipschitz-Killing curvatures of \(M\) with respect to a Riemannian metric induced on \(M\) by \(f\), and the \(\mathcal{M}_j^k(D)\) are certain Minkowski-like functionals on \(\mathbb{R}^k\) under the Gauss measure. Decompositions of this type are typical in the integral geometry, where they appear in the so-called kinematic formula. A long list of earlier (deterministic) results of this type includes also Federer’s coarea formula and Weyl’s tube formula. If \(M\) is the real line, then this decomposition turns into the famous Rice formula for uprossings of Gaussian processes.

Since the Euler-Poincaré characteristic vanishes on the empty set, it is possible to use its expected value to approximate the probability that \(A_D\) is non-empty. The second most important result of this book is a rigorous upper bound for the difference \[ | \mathbb{P}\{A_D\neq\emptyset\}-\mathbb{E} \phi(A_D)| \,. \] If \(f\) takes values in \(\mathbb{R}^1\) and \(D=[u,\infty)\), then \(A_D\neq\emptyset\) means exactly that \(f\) exceeds \(u\) for some \(t\in M\). For large \(u\), the authors provide an explicit sharp bound for the absolute difference between \(\mathbb{P}\{\sup_{t\in M}f(t)\geq u\}\) and the expected value of the Euler-Poincaré characteristic of the excursion set \(\{t\in M:\; f(t)\geq u\}\). This approach to excursion probabilities is extremely useful, since it provides an excellent approximation in a very wide setting of Gaussian fields on manifolds.

The results are then extended for non-Gaussian random fields (like \(\chi^2\) and \(F\) random fields) that can be obtained as functions of several i.i.d. Gaussian random fields, which often appear in applications. A follow-up to this book co-authored by Keith Worsley will deal with numerous applications of the presented theory.

These new developments form a core of this book, being the last of its three parts. The first two parts aim to prepare the reader for it. Part I collects basic results about Gaussian random fields, boundedness and continuity, inequalities, orthogonal expansions, excursion probabilities using entropy bounds, spectral representation and stationary fields.

Part II introduces numerous necessary geometrical facts that are vital for understanding of the new techniques to deal with random fields. Integral geometric formulae mostly deal with convex sets and their locally finite unions. In order to extend these formulae for excursion sets of smooth random fields one needs tools from differential geometry. For instance, the intrinsic volumes (or Minkowski functionals) that are defined for convex sets and their finite unions become the Lipschitz-Killing curvatures in the differential geometric setting. Part II begins with foundations of integral and differential geometry, then outlines various basic facts about piecewise smooth manifolds, critical point (Morse) theory and volume of tubes related to curvature measures and Lipschitz-Killing curvatures. Also differential geometers can find something new in this chapter, notably the kinematic formula in Gauss space, not saying about probabilistic interpretation of the tube formula and other fundamental geometric results. In Part III the authors also present a probabilistic proof of the classical Chern-Gauss-Bonnet theorem of differential geometry.

The presentation, typography and graphical illustrations are excellent, although numerous and sometimes excessively long footnotes could better join the main text.

The book is understandable for students who have covered the basic probability course with a good background in analysis. Parts of this book can be used as the core text for several courses. For instance, Part I can be used for a semester course on Gaussian processes, Part II is an excellent course on geometry for probabilists, Part III can form a base for the advances course about excursion sets for Gaussian fields.

Among numerous texts on Gaussian processes and fields, this book has a special well-identified place. It is the first monograph that covers a bulk of recent journal literature devoted to geometric properties and extremes of random fields. It can be complemented by further texts that deal with high level exceedances and extremes [M. R. Leadbetter, G. Lindgren, H. Rootzen, Extremes and related properties of random sequences and processes. Springer Series in Statistics. (New York - Heidelberg - Berlin): Springer- Verlag. (1983; Zbl 0518.60021)] and [V. I. Piterbarg, Asymptotic methods in the theory of Gaussian processes and fields. Translations of Mathematical Monographs. 148. Providence, RI: AMS. (1996; Zbl 0841.60024)], spectral theory of random fields and limit theorems [A. V. Ivanov, N. N. Leonenko, Statistical analysis of random fields. Mathematics and Its Applications: Soviet Series, 28. (Dordrecht) etc.: Kluwer Academic Publishers. (1989; Zbl 0713.62094)] and Markov fields and potential theory [D. Khoshnevisan, Multiparameter processes. An introduction to random fields. Springer Monographs in Mathematics. (New York), NY: Springer. (2002; Zbl 1005.60005)].

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.

Since the Euler-Poincaré characteristic vanishes on the empty set, it is possible to use its expected value to approximate the probability that \(A_D\) is non-empty. The second most important result of this book is a rigorous upper bound for the difference \[ | \mathbb{P}\{A_D\neq\emptyset\}-\mathbb{E} \phi(A_D)| \,. \] If \(f\) takes values in \(\mathbb{R}^1\) and \(D=[u,\infty)\), then \(A_D\neq\emptyset\) means exactly that \(f\) exceeds \(u\) for some \(t\in M\). For large \(u\), the authors provide an explicit sharp bound for the absolute difference between \(\mathbb{P}\{\sup_{t\in M}f(t)\geq u\}\) and the expected value of the Euler-Poincaré characteristic of the excursion set \(\{t\in M:\; f(t)\geq u\}\). This approach to excursion probabilities is extremely useful, since it provides an excellent approximation in a very wide setting of Gaussian fields on manifolds.

The results are then extended for non-Gaussian random fields (like \(\chi^2\) and \(F\) random fields) that can be obtained as functions of several i.i.d. Gaussian random fields, which often appear in applications. A follow-up to this book co-authored by Keith Worsley will deal with numerous applications of the presented theory.

These new developments form a core of this book, being the last of its three parts. The first two parts aim to prepare the reader for it. Part I collects basic results about Gaussian random fields, boundedness and continuity, inequalities, orthogonal expansions, excursion probabilities using entropy bounds, spectral representation and stationary fields.

Part II introduces numerous necessary geometrical facts that are vital for understanding of the new techniques to deal with random fields. Integral geometric formulae mostly deal with convex sets and their locally finite unions. In order to extend these formulae for excursion sets of smooth random fields one needs tools from differential geometry. For instance, the intrinsic volumes (or Minkowski functionals) that are defined for convex sets and their finite unions become the Lipschitz-Killing curvatures in the differential geometric setting. Part II begins with foundations of integral and differential geometry, then outlines various basic facts about piecewise smooth manifolds, critical point (Morse) theory and volume of tubes related to curvature measures and Lipschitz-Killing curvatures. Also differential geometers can find something new in this chapter, notably the kinematic formula in Gauss space, not saying about probabilistic interpretation of the tube formula and other fundamental geometric results. In Part III the authors also present a probabilistic proof of the classical Chern-Gauss-Bonnet theorem of differential geometry.

The presentation, typography and graphical illustrations are excellent, although numerous and sometimes excessively long footnotes could better join the main text.

The book is understandable for students who have covered the basic probability course with a good background in analysis. Parts of this book can be used as the core text for several courses. For instance, Part I can be used for a semester course on Gaussian processes, Part II is an excellent course on geometry for probabilists, Part III can form a base for the advances course about excursion sets for Gaussian fields.

Among numerous texts on Gaussian processes and fields, this book has a special well-identified place. It is the first monograph that covers a bulk of recent journal literature devoted to geometric properties and extremes of random fields. It can be complemented by further texts that deal with high level exceedances and extremes [M. R. Leadbetter, G. Lindgren, H. Rootzen, Extremes and related properties of random sequences and processes. Springer Series in Statistics. (New York - Heidelberg - Berlin): Springer- Verlag. (1983; Zbl 0518.60021)] and [V. I. Piterbarg, Asymptotic methods in the theory of Gaussian processes and fields. Translations of Mathematical Monographs. 148. Providence, RI: AMS. (1996; Zbl 0841.60024)], spectral theory of random fields and limit theorems [A. V. Ivanov, N. N. Leonenko, Statistical analysis of random fields. Mathematics and Its Applications: Soviet Series, 28. (Dordrecht) etc.: Kluwer Academic Publishers. (1989; Zbl 0713.62094)] and Markov fields and potential theory [D. Khoshnevisan, Multiparameter processes. An introduction to random fields. Springer Monographs in Mathematics. (New York), NY: Springer. (2002; Zbl 1005.60005)].

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.

Reviewer: Ilya S. Molchanov (Bern)

### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G15 | Gaussian processes |

60G60 | Random fields |

53Cxx | Global differential geometry |

53C65 | Integral geometry |