Gaussian random functions, Gaussian random vectors.
(Fonctions aléatoires gaussiennes, vecteurs aléatoires gaussiens.)

*(French)*Zbl 0902.60030
Montréal: Les Publications CRM. 217 p. (1997).

This monograph collects and unifies a tremendous amount of pure-mathematical research on the path behavior of Gaussian processes, at the same time presenting extensions to processes with values in Fréchet or Banach spaces. These extensions are surprisingly natural and painless, and allow statements about processes with higher-dimensional time parameter to be proved along with results for times in \({\mathbb{R}}^+\). A central technical theme of the book is that Gaussian processes with general time-parameter can be represented linearly in terms of an integer-indexed white-noise Gaussian sequence, after which general topological constructions and isoperimetric-type inequalities give qualitative information about path behavior.

The Gaussian process properties treated in the book include 0-1 laws (Chapter 1); the comparisons (Chapter 3) of tail-probabilities for process suprema \(\sup_t X_t\) against corresponding tail probabilities of processes \(Y_t\) such that \(E(X_s-X_t)^2\leq E(Y_s-Y_t)^2\) \(\forall s,t\); path-boundedness and continuity (Chapters 4, 5); and iterated-logarithm and Strassen-type asymptotic laws characterizing almost-sure functional limits of process trajectories (Chapter 6). Final chapters are devoted to the already mentioned extensions to processes with general time-parameter and to the regularity of random Fourier series with coefficient sequence invariant in law under sign-changes. The book is essentially definitive on the Gaussian-process topics it covers, which have been largely completed as research areas over the course of the past 30 years. The author has been a prime contributor to this development, especially in the 70’s, and the particular unifying approaches of this book have been foreshadowed in several of his later papers.

Other topics, involving Gaussian processes but requiring different mathematical tools, are not even touched on in the book under review: the fine structure of Brownian motion, ergodic theory of Gaussian processes, multiple Wiener-Itô stochastic integrals, and level- and curve-crossing behavior. In particular, no mention is made of non-mathematical applications of the material of the book, such as the extensive uses in engineering and statistical signal-processing of the special features of Gaussian processes. As a result, much of the underlying motivation and history have been abstracted away: for example, Slepian’s (1962) applied motivations (related to level-crossings by stationary processes) for studying comparison inequalities play no role in the book, and the elegant result (Theorem 3.2.5) which subsumes Slepian’s is not specialized to reproduce his famous lemma.

The primary tools used in the book are the theory of probability on metric spaces, isoperimetric inequalities, duality theory of Frechet and Banach spaces, reproducing kernel Hilbert spaces, metric entropy and majorizing measures. The author does a remarkably clear job of summarizing the background theory which he assumes the reader has seen elsewhere, so much so that this research monograph could successfully – if properly supplemented by classical references on these tools – be used for teaching an advanced graduate course, which would be nearly self-contained with respect to coverage of Gaussian processes. However, in a course there would need to be many more examples. Perhaps the most beautiful aspect of the book – which makes it unique, as far as the reviewer is aware – is its unified treatment of the deep and thoroughgoing connection between the reproducing kernel Hilbert space (RKHS) associated with a Gaussian process and its path behavior.

The background theory is given in Chapter 2, including a very modern (i.e., abstract) treatment of the representations by random series which were formerly called Karhunen-Loève expansions. There also is introduced the powerful isoperimetric-type inequality of Sudakov-Tsirelson-Borell, which is the primary vehicle through which the RKHS continues to reappear in the treatment of path-boundedness and continuity in later chapters. One of the technical features of Gaussian processes which makes them special among stochastic processes is the possibility of bounding \(E \sup_t X_t\) (both above and below) simply in terms of underlying process characteristics. As the author says in Chapter 5, “in all estimates regarding Gaussian processes, one remarks the phenomena of decoupling … which lead to bounds on a quantity of the type \(E \sup_t X_t\) in terms of another quantity of the type \(\int \varphi(\mu) \mu(du)\)” [where \( \mu \) is the law of the process]. The theory of majorizing measures, expounded in Chapter 5, gives the highest development of this idea, providing detailed treatment of Talagrand’s (relatively recent) solution of the long-standing problem of characterizing those (nonstationary) Gaussian processes which have continuous sample paths.

The organization of the material in the book is impeccable, and the writing is very clear, but there are some (not many) flaws in exposition, mostly related to the omission of standard examples which would motivate certain key results and render them more understandable. For instance, after a beautiful presentation of a very general result (Theorem 6.7.2), a generalization of a theorem of T. L. Lai [Z. Wahrscheinlichkeitstheorie Verw. Geb. 29, 7-19 (1974; Zbl 0272.60024)] connecting Strassen-type laws on the set of almost-sure limit points of paths in function space of a sequence of identically distributed Gaussian processes to their RKHS, the author omits explaining to the reader why the result essentially contains (as a very special case!) Strassen’s celebrated functional form of the law of the iterated logarithm for Wiener process. The book generally has rather few, but very well-chosen, examples. There are some, but relatively few, misprints. Although the book has no index, the results are presented in such a way that the key theorems in each chapter are easy to refer to. In summary, this is a beautiful book, rewarding for all researchers and advanced students in abstract probability theory who read French.

The Gaussian process properties treated in the book include 0-1 laws (Chapter 1); the comparisons (Chapter 3) of tail-probabilities for process suprema \(\sup_t X_t\) against corresponding tail probabilities of processes \(Y_t\) such that \(E(X_s-X_t)^2\leq E(Y_s-Y_t)^2\) \(\forall s,t\); path-boundedness and continuity (Chapters 4, 5); and iterated-logarithm and Strassen-type asymptotic laws characterizing almost-sure functional limits of process trajectories (Chapter 6). Final chapters are devoted to the already mentioned extensions to processes with general time-parameter and to the regularity of random Fourier series with coefficient sequence invariant in law under sign-changes. The book is essentially definitive on the Gaussian-process topics it covers, which have been largely completed as research areas over the course of the past 30 years. The author has been a prime contributor to this development, especially in the 70’s, and the particular unifying approaches of this book have been foreshadowed in several of his later papers.

Other topics, involving Gaussian processes but requiring different mathematical tools, are not even touched on in the book under review: the fine structure of Brownian motion, ergodic theory of Gaussian processes, multiple Wiener-Itô stochastic integrals, and level- and curve-crossing behavior. In particular, no mention is made of non-mathematical applications of the material of the book, such as the extensive uses in engineering and statistical signal-processing of the special features of Gaussian processes. As a result, much of the underlying motivation and history have been abstracted away: for example, Slepian’s (1962) applied motivations (related to level-crossings by stationary processes) for studying comparison inequalities play no role in the book, and the elegant result (Theorem 3.2.5) which subsumes Slepian’s is not specialized to reproduce his famous lemma.

The primary tools used in the book are the theory of probability on metric spaces, isoperimetric inequalities, duality theory of Frechet and Banach spaces, reproducing kernel Hilbert spaces, metric entropy and majorizing measures. The author does a remarkably clear job of summarizing the background theory which he assumes the reader has seen elsewhere, so much so that this research monograph could successfully – if properly supplemented by classical references on these tools – be used for teaching an advanced graduate course, which would be nearly self-contained with respect to coverage of Gaussian processes. However, in a course there would need to be many more examples. Perhaps the most beautiful aspect of the book – which makes it unique, as far as the reviewer is aware – is its unified treatment of the deep and thoroughgoing connection between the reproducing kernel Hilbert space (RKHS) associated with a Gaussian process and its path behavior.

The background theory is given in Chapter 2, including a very modern (i.e., abstract) treatment of the representations by random series which were formerly called Karhunen-Loève expansions. There also is introduced the powerful isoperimetric-type inequality of Sudakov-Tsirelson-Borell, which is the primary vehicle through which the RKHS continues to reappear in the treatment of path-boundedness and continuity in later chapters. One of the technical features of Gaussian processes which makes them special among stochastic processes is the possibility of bounding \(E \sup_t X_t\) (both above and below) simply in terms of underlying process characteristics. As the author says in Chapter 5, “in all estimates regarding Gaussian processes, one remarks the phenomena of decoupling … which lead to bounds on a quantity of the type \(E \sup_t X_t\) in terms of another quantity of the type \(\int \varphi(\mu) \mu(du)\)” [where \( \mu \) is the law of the process]. The theory of majorizing measures, expounded in Chapter 5, gives the highest development of this idea, providing detailed treatment of Talagrand’s (relatively recent) solution of the long-standing problem of characterizing those (nonstationary) Gaussian processes which have continuous sample paths.

The organization of the material in the book is impeccable, and the writing is very clear, but there are some (not many) flaws in exposition, mostly related to the omission of standard examples which would motivate certain key results and render them more understandable. For instance, after a beautiful presentation of a very general result (Theorem 6.7.2), a generalization of a theorem of T. L. Lai [Z. Wahrscheinlichkeitstheorie Verw. Geb. 29, 7-19 (1974; Zbl 0272.60024)] connecting Strassen-type laws on the set of almost-sure limit points of paths in function space of a sequence of identically distributed Gaussian processes to their RKHS, the author omits explaining to the reader why the result essentially contains (as a very special case!) Strassen’s celebrated functional form of the law of the iterated logarithm for Wiener process. The book generally has rather few, but very well-chosen, examples. There are some, but relatively few, misprints. Although the book has no index, the results are presented in such a way that the key theorems in each chapter are easy to refer to. In summary, this is a beautiful book, rewarding for all researchers and advanced students in abstract probability theory who read French.

Reviewer: E.Slud (College Park)

##### MSC:

60G15 | Gaussian processes |

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

60G17 | Sample path properties |

60B11 | Probability theory on linear topological spaces |