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This book provides an in-depth account of modern methods used to bound the supremum of stochastic processes. Starting from first principles, it takes the reader to the frontier of current research. This second edition has been completely rewritten, offering substantial improvements to the exposition and simplified proofs, as well as new results.
The book starts with a thorough account of the generic chaining, a remarkably simple and powerful method to bound a stochastic process that should belong to every probabilist’s toolkit. The effectiveness of the scheme is demonstrated by the characterization of sample boundedness of Gaussian processes. Much of the book is devoted to exploring the wealth of ideas and results generated by thirty years of efforts to extend this result to more general classes of processes, culminating in the recent solution of several key conjectures.
A large part of this unique book is devoted to the author’s influential work. While many of the results presented are rather advanced, others bear on the very foundations of probability theory. In addition to providing an invaluable reference for researchers, the book should therefore also be of interest to a wide range of readers.
What Is This Book About?
The Generic Chaining
Gaussian Processes and the Generic Chaining
Trees and Other Measures of Size
Matching Theorems
Some Dreams Come True
Warming Up with p-Stable Processes.
Bernoulli Processes
Random Fourier Series and Trigonometric Sums
Partitioning Scheme and Families of Distances
Peaky Part of Functions
Proof of the Bernoulli Conjecture
Random Series of Functions
Infinitely Divisible Processes
Unfulfilled Dreams
Practicing
Empirical Processes, II.
Gaussian Chaos
Convergence of Orthogonal Series: Majorizing Measures
Shor’s Matching Theorem
The Ultimate Matching Theorem in Dimension 3
Applications to Banach Space Theory
A Discrepancy for Convex Sets
B Some Deterministic Arguments
C Classical View of Infinitely Divisible Processes
D Reading Suggestions
E Research Directions
F Solutions of Selected Exercises
G Comparison with the First Edition