In addition to its central focus on generic chaining, which allows for optimal bounds in Gaussian and Bernoulli processes, this volume on modern stochastic methods includes key applications and a variety of complete solutions to a number of classical problems.

The book develops modern methods and in particular the "generic chaining" to bound stochastic processes. This methods allows in particular to get optimal bounds for Gaussian and Bernoulli processes. Applications are given to stable processes, infinitely divisible processes, matching theorems, the convergence of random Fourier series, of orthogonal series, and to functional analysis. The complete solution of a number of classical problems is given in complete detail, and an ambitious program for future research is laid out.

Autorentext

Michel Talagrand has made profound contributions to mathematics, notably in probability theory and related topics. The author of several books and well over 200 research papers, he is the recipient of several awards, including the Loève Prize, the Fermat Prize and the Shaw Prize. He has been a plenary speaker at the International Congress of Mathematicians and is a member of the French Academy of Sciences.

Inhalt

0. Introduction.- 1. Philosophy and Overview of the Book.- 2. Gaussian Processes and the Generic Chaining.- 3. Random Fourier Series and Trigonometric Sums, I. - 4. Matching Theorems I.- 5. Bernouilli Processes.- 6. Trees and the Art of Lower Bounds.- 7. Random Fourier Series and Trigonometric Sums, II.- 8. Processes Related to Gaussian Processes.- 9. Theory and Practice of Empirical Processes.- 10. Partition Scheme for Families of Distances.- 11. Infinitely Divisible Processes.- 12. The Fundamental Conjectures.- 13. Convergence of Orthogonal Series; Majorizing Measures.- 14. Matching Theorems, II: Shor's Matching Theorem. 15. The Ultimate Matching Theorem in Dimension 3.- 16. Applications to Banach Space Theory.- 17. Appendix: What this Book is Really About.- 18. Appendix: Continuity.- References. Index.