Talagrand M. Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems

B: Springer, 2014. - 626p.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.Philosophy and Overview of the Book.
Gaussian Processes and the Generic Chaining.
Random Fourier Series and Trigonometric Sums, I.
Matching Theorems, I.
Bernoulli Processes.
Trees and the Art of Lower Bounds.
Random Fourier Series and Trigonometric Sums, II.
Processes Related to Gaussian Processes.
Theory and Practice of Empirical Processes.
Partition Scheme for Families of Distances.
Infinitely Divisible Processes.Pages 331-370.
The Fundamental Conjectures.Pages 371-398.
Convergence of Orthogonal Series; Majorizing Measures.
Matching Theorems, II: Shor’s Matching Theorem.
The Ultimate Matching Theorem in Dimension >3.
Applications to Banach Space Theory.