×

Gaussian random functions. (English) Zbl 0832.60002

Mathematics and its Applications (Dordrecht). 322. Dordrecht: Kluwer Academic Publishers. xi, 333 p. (1995).
The theory of Gaussian random functions during the last three decades was one of the most advanced fields of the probability science attracting attention of many leading probabilists of our time. This can be explained by several factors: it is well-known what role plays Gaussian measure in infinite-dimensional spaces, where there is no analogue of the Lebesgue measure; the new important effects connected with infinite dimensionality were discovered; methods initially developed for Gaussian measures were found useful in other subjects of mathematics, especially in functional analysis. Therefore it is not surprising that during the last decade there appeared several books and survey papers on different aspects of the theory of Gaussian random functions. The book under consideration, written by the representative of strong St. Petersburg probability school, is one of them.
As it is written in the preface of the book, it is intended as a textbook, therefore the author focuses on quite a few fundamental objects in the theory and tries to discover their interrelations. By concentrating on the principal points, the author does not aim at covering all the material available by now. The following topics are treated in the book: the kernel of a Gaussian measure, the model of a Gaussian random function, oscillations of sample functions, the convexity and isoperimetric inequalities, regularity of sample functions, functional laws of the iterated logarithm, estimates for the probabilities of large deviations, small deviations problem. The last two topics are the ones to which the author’s personal contribution is the most significant.
We shall not give a detailed description of the results of each section; instead of that we would like to mention that the author systematically uses the very important notion of convexity of measures, metric entropy, majorizing measures and isoperimetric inequalities. New and more simple proofs (which appeared during the last years) of a number of fundamental results, such as necessary and sufficient conditions for boundedness and continuity of Gaussian random functions, are presented in the book. Also I think that the author’s comments on the historical reference to the origins of the results of the book and the recommendations for further study on related problems will be of great use for a reader; as an example one can mention the observation (originally due to V. N. Sudakov) that the well-known Slepian inequality is a probabilistic interpretation of the classical Schäfli theorem, proved more than one hundred years ago. At the end of the book eleven open problems are formulated. A comprehensive list of references, containing more than 400 items reflects well the literature in Russian, English and French.
Concluding I can express my firm belief that this book will be of interest both for specialists working in the area and for students who want to enter this interesting field of research.

MSC:

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60G50 Sums of independent random variables; random walks
60F10 Large deviations
PDFBibTeX XMLCite