The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

* Recasts topics in random fields by following a completely new way of handling both geometry and probability
 * Significant exposition of the work of others in the field
 * Presentation is clear and pedagogical
 * Excellent reference work as well as excellent work for self study

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.

The three parts to the monograph are quite distinct. Part I presents a user-friendly yet comprehensive background to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities. Part II gives a quick review of geometry, both integral and Riemannian, to provide the reader with the material needed for Part III, and to give some new results and new proofs of known results along the way. Topics such as Crofton formulae, curvature measures for stratified manifolds, critical point theory, and tube formulae are covered. In fact, this is the only concise, self-contained treatment of all of the above topics, which are necessary for the study of random fields. The new approach in Part III is devoted to the geometry of excursion sets of random fields and the related Euler characteristic approach to extremal probabilities.

"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. These applications, to appear in a forthcoming volume, will cover areas as widespread as brain imaging, physical oceanography, and astrophysics.

Random Fields and Geometry

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrodinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.

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Physical Mechanisms of the Rogue Wave Phenomenon

Linear models

This chapter assesses the causes of observed changes assessed in Chapters 2 to 5 and uses understanding of physical processes, climate models and statistical approaches. The chapter adopts the terminology for detection and attribution proposed by the IPCC good practice guidance paper on detection and attribution (Hegerl et al., 2010) and for uncertainty Mastrandrea et al. (2011). Detection and attribution of impacts of climate changes are assessed by Working Group II, where Chapter 18 assesses the extent to which atmospheric and oceanic changes influence ecosystems, infrastructure, human health and activities in economic sectors.

Detection and attribution of climate change: from global to regional

Introduction. Reading diagram. Chapter 1: Classical results on the regularity of the paths. 1. Kolmogorov's Extension Theorem. 2. Reminder on the Normal Distribution. 3. 0-1 law for Gaussian processes. 4. Regularity of the paths. Exercises. Chapter 2: Basic Inequalities for Gaussian Processes. 1. Slepian type inequalities. 2. Ehrhard's inequality. 3. Gaussian isoperimetric inequality. 4. Inequalities for the tails of the distribution of the supremum. 5. Dudley's inequality. Exercises. Chapter 3: Crossings and Rice formulas for 1-dimensional parameter processes. 1. Rice Formulas. 2. Variants and Examples. Exercises. Chapter 4: Some Statistical Applications. 1. Elementary bounds for P{M > u}. 2. More detailed computation of the first two moments. 3. Maximum of the absolute value. 4. Application to quantitative gene detection. 5. Mixtures of Gaussian distributions. Exercises. Chapter 5: The Rice Series. 1. The Rice Series. 2. Computation of Moments. 3. Numerical aspects of Rice Series. 4. Processes with Continuous Paths. Chapter 6: Rice formulas for random fields. 1. Random fields from Rd to Rd. 2. Random fields from Rd to Rd!, d> d!. Exercises. Chapter 7: Regularity of the Distribution of the Maximum. 1. The implicit formula for the density of the maximum. 2. One parameter processes. 3. Continuity of the density of the maximum of random fields. Exercises. Chapter 8: The tail of the distribution of the maximum. 1. One-dimensional parameter: asymptotic behavior of the derivatives of FM. 2. An Application to Unbounded Processes. 3. A general bound for pM. 4. Computing p(x) for stationary isotropic Gaussian fields. 5. Asymptotics as x! +". 6. Examples. Exercises. Chapter 9: The record method. 1. Smooth processes with one dimensional parameter. 2. Non-smooth Gaussian processes. 3. Two-parameter Gaussian processes. Exercises. Chapter 10: Asymptotic methods for infinite time horizon. 1. Poisson character of "high" up-crossings. 2. Central limit theorem for non-linear functionals. Exercises. Chapter 11: Geometric characteristics of random sea-waves. 1. Gaussian model for infinitely deep sea. 2. Some geometric characteristics of waves. 3. Level curves, crests and velocities for space waves. 4. Real Data. 5. Generalizations of the Gaussian model. Exercises. Chapter 12: Systems of random equations. 1. The Shub-Smale model. 2. More general models. 3. Non-centered systems (smoothed analysis). 4. Systems having a law invariant under orthogonal transformations and translations. Chapter 13: Random fields and condition numbers of random matrices. 1. Condition numbers of non-Gaussian matrices. 2. Condition numbers of centered Gaussian matrices. 3. Non-centered Gaussian matrices. Notations. References.

Level sets and extrema of random processes and fields

Spike detection from inaccurate samplings

We consider designs in linear blocks with border plots, in which a treatment may affect the response on the two adjacent plots. Three series of designs are given: (i) neighbour-balanced designs in complete blocks; (ii) neighbour-balanced designs in blocks each of which lacks one treatment; (iii) partially neighbour-balanced designs in few complete blocks. Complete methods of construction and randomization are given. Optimality properties are discussed.

A catalogue of efficient neighbour-designs with border plots

This article investigates the support detection problem using the LASSO estimator in the space of measures. More precisely, we study the recovery of a discrete measure (spike train) from few noisy observations (Fourier samples, moments...) using an $\ell_{1}$-regularization procedure. In particular, we provide an explicit quantitative localization of the spikes.

The “optimal fingerprint” method, usually used for detection and attribution studies, requires to know, or, in practice, to estimate the covariance matrix of the internal climate variability. In this work, a new adaptation of the “optimal fingerprints” method is presented. The main goal is to allow the use of a covariance matrix estimate based on an observation dataset in which the number of years used for covariance estimation is close to the number of observed time series. Our adaptation is based on the use of a regularized estimate of the covariance matrix, that is well-conditioned, and asymptotically more precise, in the sense of the mean square error. This method is shown to be more powerful than the basic “guess pattern fingerprint”, and than the classical use of a pseudo-inverted truncation of the empirical covariance matrix. The construction of the detection test is achieved by using a bootstrap technique particularly well-suited to estimate the internal climate variability in real world observations. In order to validate the efficiency of the detection algorithm with climate data, the methodology presented here is first applied with pseudo-observations derived from transient regional climate change scenarios covering the 1960-2099 period. It is then used to perform a formal detection study of climate change over France, analyzing homogenized observed temperature series from 1900 to 2006. In this case, the estimation of the covariance matrix is only based on a part of the observation dataset. This new approach allows the confirmation and extension of previous results regarding the detection of an anthropogenic climate change signal over the country.

Jean-Marc Azaïs

Papers

Level sets and extrema of random processes and fields

Spike detection from inaccurate samplings

A catalogue of efficient neighbour-designs with border plots

Spike detection from inaccurate samplings

Adaptation of the optimal fingerprint method for climate change detection using a well-conditioned covariance matrix estimate