Some random series of functions. 2. ed. (English) Zbl 0805.60007

Kahane, Jean-Pierre

Cambridge Studies in Advanced Mathematics. 5. Cambridge: Cambridge University Press. xiii, 305 p. (1993).

The first edition of this book in 1968 (Zbl 0192.538) was very influential no doubt due to its clarity of exposition and the many interesting problems that it posed. The second edition differs from the first one by two completely rewritten chapters and three new chapters. The text in the other chapters is almost unchanged with some new sections added. Therefore we will refer only new or revised material in the second edition of the book.
Chapter 10, entitled “A few geometrical notions”, serves as introduction to the second half of the book. It contains the definition of the Hausdorff dimension and Frostman’s theory of Hausdorff measures. \(\varepsilon\)-covering numbers, quasi-helices, von Koch and Assoud curves are discussed as well.
Chapter 11 deals with random translates and covering. In its first part the random set consisting of the points of the circle that are covered infinitely often by open intervals (= arcs) of length smaller than one, distributed at random on the circle, is considered. In the second part for a torus \(T^ q\), \(q\geq 1\), and a Borelian subset \(A\subset T^ q\) the problem of finding necessary and sufficient conditions for \(A\subset \varlimsup {\mathcal G}_ n\) a.s., where \({\mathcal G}_ i\), \(i=1,2, \dots,\) are random translates of a sequence of some open subsets of \(T^ q\), is solved.
Chapter 15 is devoted to a.s. boundedness and continuity for Gaussian processes and contains Slepian’s lemma, Marcus’ and Shepp’s theorem with application to the Pisier algebra, the theorems of Dudley and Fernique and similar problems for non-Gaussian Fourier series. Chapter 16 deals with expression in the Fourier series of the Brownian motion (real or complex) and its sample properties. In Chapter 18 the structure of fractional Brownian images and level sets is investigated, their Hausdorff dimension estimated, the relation with the occupation density employed.
Chapter 12, Section 7 contains Gaussian series in Banach space. In Chapter 6 sections on polynomials with unimodular coefficients and on sums of sinuses are new. To Chapter 5 a section on Fourier coefficients of continuous functions and to Chapter 2 one on strong integrability for Rademacher series are added. Chapter 13 contains a section on the range of a Gaussian Taylor series in the unit circle.

Reviewer: N.Kalinauskaitė (Vilnius)

MSC:

60B99	Probability theory on algebraic and topological structures
60-02	Research exposition (monographs, survey articles) pertaining to probability theory
40-02	Research exposition (monographs, survey articles) pertaining to sequences, series, summability

Keywords:

Hausdorff dimension; Hausdorff measures; Pisier algebra; structure of fractional Brownian images; Rademacher series; Gaussian Taylor series

Citations:

Zbl 0571.60002; Zbl 0192.538