Michel Talagrand

Upper and Lower Bounds for Stochastic Processes

Decomposition Theorems

• Gebundenes Buch

Produktbeschreibung

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.

Produktdetails

• Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
• Verlag: Springer / Springer International Publishing / Springer, Berlin
• Artikelnr. des Verlages: 978-3-030-82594-2
• 2. Aufl.
• Seitenzahl: 744
• Erscheinungstermin: 21. Dezember 2021
• Englisch
• Abmessung: 241mm x 160mm x 46mm
• Gewicht: 1267g
• ISBN-13: 9783030825942
• ISBN-10: 3030825949
• Artikelnr.: 62148370

Autorenporträt

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.

Inhaltsangabe

1. What is This Book About? Part I The Generic Chaining.- 2 Gaussian Processes and the Generic Chaining.- 3 Trees and Other Measures of Size.- 4 Matching Theorems.- Part II Some Dreams Come True.- 5 Warming Up with p-Stable Processes.- 6 Bernoulli Processes.- 7 Random Fourier Series and Trigonometric Sums.- 8 Partitioning Scheme and Families of Distances.- 9 Peaky Part of Functions.- 10 Proof of the Bernoulli Conjecture.- 11 Random Series of Functions.- 12 Infinitely Divisible Processes.- 13 Unfulfilled Dreams.- Part III Practicing.- 14 Empirical Processes, II.- 15 Gaussian Chaos.- 16 Convergence of Orthogonal Series; Majorizing Measures.- 17 Shor's Matching Theorem.- 18 The Ultimate Matching Theorem in Dimension Three.- 19 Application 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 withthe First Edition.- References.- Index.

Rezensionen

"The book includes a rich collection of exercises that will allow the reader to gain skills for a better understanding. The book is then suitable as a textbook for an advanced course. ... The systematic and deep treatment of the subject under study makes the book a good reference for the specialist." (Erick Treviño-Aguilar, Mathematical Reviews, March, 2023)

"Although he writes a book about inequalities of stochastic processes, Talagrand focuses on modern abstract methods, completely abdicating the 'classical approach'. ... Talagrand's goal in this book is, without any doubt, very ambitious. ... it contains marvelous ideas that should very likely be in the toolbox of anyone dealing with stochastic processes." (Antonio Auffinger, Bulletin of the American Mathematical Society, Vol. 53 (1), January, 2016)

"Each chapter begins with an explanation of the overall philosophy and gives a brief survey of the things to come. This makes a rewarding read for everyone with a sound probability background, and for the specialist it is even entertaining and most inspiring. ... There are many areas where Talagrand has a very personal view of things which, I am sure, will be food for thought for future generations of probabilists."(René L. Schilling, The Mathematical Gazette, Vol. 100 (547), 2016)
"The topic of this book is the study of the supremum of certain stochastic processes, more precisely, it describes how to find upper and lower bounds for this suprema. ... The book can be interesting for all potential readers who study the modern stochastic methods, for undergraduate and postgraduate students, for specialists in the theory of stochastic processes and for practitioners who treat the trajectories of stochastic models." (Yuliya S. Mishura, zbMATH, Vol. 1293, 2014)