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**Probability in Banach spaces. Isoperimetry and processes.**
*(English)*
Zbl 0748.60004

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 23. Berlin etc.: Springer-Verlag. xii, 480 p. (1991).

As the authors state, “this book tries to present some of the main aspects of the theory of probability in Banach spaces, from the foundation of the topic to the latest developments and current research questions”. The authors have succeeded admirably; the book sums up and discusses most of the important results of probability theory in Banach spaces which have been discovered in the last two decades. Probability in Banach spaces is a branch of mathematics that lies in a joint of classical probability, measure theory and functional analysis; it studies properties of random variables taking values in a Banach space, the behaviour of their distributions, limit theorems etc. These properties as well as statements of the limit theorems are essentially dependent on the geometry of the Banach space, and this fact stipulates the nature of the theory, where the methods of classical probability interlace with those of measure theory, geometry of a Banach space and with abstract analytic methods.

This very comprehensive book develops a wide variety of the methods existing nowadays in probability in Banach spaces. However the authors select among them and focus the reader’s attention on the isoperimetric inequalities method, concentration of measure phenomenon, and the abstract random processes technique. This is reflected in the subtitle of the book: Isoperimetry and processes. The authors themselves contributed essentially to the elaboration of these methods. The use of isoperimetric inequalities to the concentration inequalities, tail estimates and integrability theorems of probability in Banach spaces has led today to rather a complete picture of the theory. The second part of the subtitle – Processes – is related to a big chapter of probability in Banach spaces to which the second author has contributed very much. Here the central issue is the celebrated Fernique-Talagrand theorem solving the problem on continuity and boundedness of samples of a Gaussian process by means of the majorizing measure technique. This material is naturally included into the framework of probability in Banach spaces, since for instance, a random process with continuous sample paths can be considered as a random variable taking values in the Banach space of continuous functions.

It seems to use that the monograph will become an event for mathematicians working or interested in probability in Banach spaces. The table of contents we bring below shows that the monograph covers most of the main questions of the theory. The authors did not intend to cover all the aspects of the theory. Among the topics not covered they mention Banach space valued martingales, infinitely divisible distributions, rate of convergence in the central limit theorem. We would also add characteristic functionals, their topological descriptions, cylindrical measures and Radonifying operators. The book is equipped with an extensive bibliography.

Table of contents: Chapter 1. Isoperimetric inequalities and the concentration of measure phenomenon; Chapter 2. Generalities on Banach space valued random variables and random processes; Chapter 3. Gaussian random variables; Chapter 4. Rademacher averages; Chapter 5. Stable random variables; Chapter 6. Sums of independent random variables; Chapter 7. The strong law of large numbers; Chapter 8. The law of iterated logarithm; Chapter 9. Type and cotype of Banach spaces; Chapter 10. The central limit theorem; Chapter 11. Regularity of random processes; Chapter 12. Regularity of Gaussian and stable processes; Chapter 13. Stationary processes and random Fourier series; Chapter 14. Empirical process methods in probability in Banach spaces; Chapter 15. Applications to Banach space theory.

This very comprehensive book develops a wide variety of the methods existing nowadays in probability in Banach spaces. However the authors select among them and focus the reader’s attention on the isoperimetric inequalities method, concentration of measure phenomenon, and the abstract random processes technique. This is reflected in the subtitle of the book: Isoperimetry and processes. The authors themselves contributed essentially to the elaboration of these methods. The use of isoperimetric inequalities to the concentration inequalities, tail estimates and integrability theorems of probability in Banach spaces has led today to rather a complete picture of the theory. The second part of the subtitle – Processes – is related to a big chapter of probability in Banach spaces to which the second author has contributed very much. Here the central issue is the celebrated Fernique-Talagrand theorem solving the problem on continuity and boundedness of samples of a Gaussian process by means of the majorizing measure technique. This material is naturally included into the framework of probability in Banach spaces, since for instance, a random process with continuous sample paths can be considered as a random variable taking values in the Banach space of continuous functions.

It seems to use that the monograph will become an event for mathematicians working or interested in probability in Banach spaces. The table of contents we bring below shows that the monograph covers most of the main questions of the theory. The authors did not intend to cover all the aspects of the theory. Among the topics not covered they mention Banach space valued martingales, infinitely divisible distributions, rate of convergence in the central limit theorem. We would also add characteristic functionals, their topological descriptions, cylindrical measures and Radonifying operators. The book is equipped with an extensive bibliography.

Table of contents: Chapter 1. Isoperimetric inequalities and the concentration of measure phenomenon; Chapter 2. Generalities on Banach space valued random variables and random processes; Chapter 3. Gaussian random variables; Chapter 4. Rademacher averages; Chapter 5. Stable random variables; Chapter 6. Sums of independent random variables; Chapter 7. The strong law of large numbers; Chapter 8. The law of iterated logarithm; Chapter 9. Type and cotype of Banach spaces; Chapter 10. The central limit theorem; Chapter 11. Regularity of random processes; Chapter 12. Regularity of Gaussian and stable processes; Chapter 13. Stationary processes and random Fourier series; Chapter 14. Empirical process methods in probability in Banach spaces; Chapter 15. Applications to Banach space theory.

Reviewer: S.A.Chobanjan (Tbilisi)

### MSC:

60Bxx | Probability theory on algebraic and topological structures |

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

60G15 | Gaussian processes |

60G17 | Sample path properties |

60G50 | Sums of independent random variables; random walks |

46Bxx | Normed linear spaces and Banach spaces; Banach lattices |

28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |

49R50 | Variational methods for eigenvalues of operators (MSC2000) |

52A40 | Inequalities and extremum problems involving convexity in convex geometry |

62G30 | Order statistics; empirical distribution functions |

60E15 | Inequalities; stochastic orderings |