Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics Book 60)

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. -- Source other Library of Congress Upper and Lower Bounds for Stochastic Processes 2 Preface 5 References 8 Contents 9 A. Appendix: What This Book Is Really About 14 B. Appendix: Continuity 26 1. Philosophy and Overview of the Book 45 1.1 Underlying Philosophy 45 1.2 Peculiarities of Style 45 1.3 What This Book Is Really About 46 1.4 Gaussian Processes and the Generic Chaining 47 1.5 Random Fourier Series and Trigonometric Sums, I 49 1.6 Matching Theorems, I 50 1.7 Bernoulli Processes 51 1.8 Trees and the Art of Lower Bounds 51 1.9 Random Fourier Series and Trigonometric Sums, II 52 1.10 Processes Related to Gaussian Processes 52 1.11 Theory and Practice of Empirical Processes 53 1.12 Partition Scheme for Families of Distances 53 1.13 Infinitely Divisible Processes 54 1.14 The Fundamental Conjectures 54 1.15 Convergence of Orthogonal Series; Majorizing Measures 54 1.16 Matching Theorems II: Shor's Matching Theorem 55 1.17 The Ultimate Matching Theorem in Dimension 3 55 1.18 Applications to Banach Space Theory 55 1.19 Appendix B: Continuity 56 Reference 56 2. Gaussian Processes and the Generic Chaining 57 3. Random Fourier Series and Trigonometric Sums, I 118 4. Matching Theorems, I 133 5. Bernoulli Processes 170 6. Trees and the Art of Lower Bounds 213 7. Random Fourier Series and Trigonometric Sums, II 238 8. Processes Related to Gaussian Processes 272 9. Theory and Practice of Empirical Processes 309 10. Partition Scheme for Families of Distances 350 11. Infinitely Divisible Processes 368 12. The Fundamental Conjectures 408 13. Convergence of Orthogonal Series; Majorizing Measures 436 14. Matching Theorems, II: Shor's Matching Theorem 484 15. The Ultimate Matching Theorem in Dimension 3 512 16. Applications to Banach Space Theory 551 Author's Note: The material of this book has been reworked and expanded with a lot more detail and published in the author's 2014 book'Upper and Lower Bounds for Stochastic Processes'(Ergebnisse Vol. 60, ISBN 978-3-642-54074-5). That book is much easier to read and covers everything that is in'The Generic Chaining'book in a more detailed and comprehensible way. ••••••••••••What is the maximum level a certain river is likely to reach over the next 25 years? (Having experienced three times a few feet of water in my house, I feel a keen personal interest in this question.) There are many questions of the same nature: what is the likely magnitude of the strongest earthquake to occur during the life of a planned building, or the speed of the strongest wind a suspension bridge will have to stand? All these situations can be modeled in the same manner. The value X of the quantity of interest (be it water t level or speed of wind) at time t is a random variable. What can be said about the maximum value of X over a certain range of t? t A collection of random variables (X), where t belongs to a certain index t set T, is called a stochastic process, and the topic of this book is the study of the supremum of certain stochastic processes, and more precisely to?nd upper and lower bounds for the quantity EsupX. (0. 1) t t?T Since T might be uncountable, some care has to be taken to de?ne this quantity. For any reasonable de?nition of Esup X we have t t?T EsupX =sup{EsupX ; F?T,F?nite}, (0. 2) t t t?T t?F an equality that we will take as the de?nition of the quantity Esup X. t t?T Thus, the crucial case for the estimation of the quantity (0. "New isoperimetric inequalities and random process techniques have recently appeared at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (e.g. boundedness and continuity of random processes, integrability and limit theorems for vector valued random variables) and of some of their links to Geometry of Banach spaces. Its purpose is to present some of the main aspects of this theory, from the foundations to the latest developments, treated with the most recent and updated tools. In particular, the most important features are the systematic use of isoperimetry and related concentration of measure phenomena (to study integrability and limit theorems for vector valued random variables), and recent abstract random process techniques (entropy and majorizing measures). Some examples of these probabilistic ideas to classical Banach space theory complete this exposition."--pub. desc

new Isoperimetric Inequalities And Random Process Techniques Have Recently Appeared At The Basis Of The Modern Understanding Of Probability In Banach Spaces. Based On These Tools, The Book Presents A Complete Treatment Of The Main Aspects Of Probability In Banach Spaces (e.g. Boundedness And Continuity Of Random Processes, Integrability And Limit Theorems For Vector Valued Random Variables) And Of Some Of Their Links To Geometry Of Banach Spaces. Its Purpose Is To Present Some Of The Main Aspects Of This Theory, From The Foundations To The Latest Developments, Treated With The Most Recent And Updated Tools. In Particular, The Most Important Features Are The Systematic Use Of Isoperimetry And Related Concentration Of Measure Phenomena (to Study Integrability And Limit Theorems For Vector Valued Random Variables), And Recent Abstract Random Process Techniques (entropy And Majorizing Measures). Some Examples Of These Probabilistic Ideas To Classical Banach Space Theory Complete This Exposition.

Front Matter....Pages I-XV Philosophy and Overview of the Book....Pages 1-12 Gaussian Processes and the Generic Chaining....Pages 13-73 Random Fourier Series and Trigonometric Sums, I....Pages 75-89 Matching Theorems, I....Pages 91-127 Bernoulli Processes....Pages 129-171 Trees and the Art of Lower Bounds....Pages 173-197 Random Fourier Series and Trigonometric Sums, II....Pages 199-232 Processes Related to Gaussian Processes....Pages 233-269 Theory and Practice of Empirical Processes....Pages 271-311 Partition Scheme for Families of Distances....Pages 313-330 Infinitely Divisible Processes....Pages 331-370 The Fundamental Conjectures....Pages 371-398 Convergence of Orthogonal Series; Majorizing Measures....Pages 399-446 Matching Theorems, II: Shor’s Matching Theorem....Pages 447-474 The Ultimate Matching Theorem in Dimension ≥3....Pages 475-513 Applications to Banach Space Theory....Pages 515-593 Back Matter....Pages 595-626 Isoperimetric, measure concentration and random process techniques appear at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (integrability and limit theorems for vector valued random variables, boundedness and continuity of random processes) and of some of their links to Geometry of Banach spaces (via the type and cotype properties). Its purpose is to present some of the main aspects of this theory, from the foundations to the most important achievements. The main features of the investigation are the systematic use of isoperimetry and concentration of measure and abstract random process techniques (entropy and majorizing measures). Examples of these probabilistic tools and ideas to classical Banach space theory are further developed. "The fundamental question of characterizing continuity and boundedness of Gaussian processes goes back to Kolmogorov. After essential contributions by R. Dudley and X. Fernique, it was solved by the author in 1985. This advance was followed by a great improvement of our understanding of the boundedness of other fundamental classes of processes (empirical processes, infinitely divisible processes, etc.). This challenging body of work has now been considerably simplified through the notion of "generic chaining", a completely natural variation on the ideas of Kolmogorov. The entirely new presentation adopted here takes the reader from the first principles to the edge of current knowledge, and to the wonderful open problems that remain in this domain."--Jacket

the Fundamental Question Of Characterizing Continuity And Boundedness Of Gaussian Processes Goes Back To Kolmogorov. After Essential Contributions By R. Dudley And X. Fernique, It Was Solved By The Author In 1985. This Advance Was Followed By A Great Improvement Of Our Understanding Of The Boundedness Of Other Fundamental Classes Of Processes (empirical Processes, Infinitely Divisible Processes, Etc.) This Challenging Body Of Work Has Now Been Considerably Simplified Through The Notion Of Generic Chaining, A Completely Natural Variation On The Ideas Of Kolmogorov. The Entirely New Presentation Adopted Here Takes The Reader From The First Principles To The Edge Of Current Knowledge, And To The Wonderful Open Problems That Remain In This Domain.

1.13 Infinitely Divisible Processes1.14 The Fundamental Conjectures; 1.15 Convergence of Orthogonal Series; Majorizing Measures; 1.16 Matching Theorems II: Shor's Matching Theorem; 1.17 The Ultimate Matching Theorem in Dimension 3; 1.18 Applications to Banach Space Theory; 1.19 Appendix B: Continuity; Reference; 2. Gaussian Processes and the Generic Chaining; 2.1 Overview; 2.2 The Generic Chaining; 2.3 Functionals; 2.4 Gaussian Processes and the Mysteries of Hilbert Space; 2.5 A First Look at Ellipsoids; 2.6 Proof of the Fundamental Partitioning Theorem; 2.7 A General Partitioning Scheme 8. Processes Related to Gaussian Processes8.1 p-Stable Processes; 8.2 Order 2 Gaussian Chaos; 8.3 Tails of Multiple Order Gaussian Chaos; 8.4 Notes and Comments; References; 9. Theory and Practice of Empirical Processes; 9.1 Discrepancy Bounds; 9.2 How to Approach Practical Problems; 9.3 The Class of Squares of a Given Class; 9.4 When Not to Use Chaining; 9.5 Notes and Comments; References; 10. Partition Scheme for Families of Distances; 10.1 The Partition Scheme; 10.2 The Structure of Certain Canonical Processes; References; 11. Infinitely Divisible Processes; 11.1 A Well-Kept Secret 2.8 Notes and CommentsReferences; 3. Random Fourier Series and Trigonometric Sums, I; 3.1 Translation Invariant Distances; 3.2 The Marcus-Pisier Theorem; 3.3 A Theorem of Fernique; 3.4 Notes and Comments; References; 4. Matching Theorems, I; 4.1 The Ellipsoid Theorem; 4.2 Matchings; 4.3 The Ajtai, Komlós, Tusnády Matching Theorem; 4.4 The Leighton-Shor Grid Matching Theorem; 4.5 Notes and Comments; References; 5. Bernoulli Processes; 5.1 Boundedness of Bernoulli Processes; 5.2 Chaining for Bernoulli Processes; 5.3 Fundamental Tools for Bernoulli Processes; 5.4 Control in Norm 5.5 Latała's Principle5.6 Chopping Maps and Functionals; 5.7 The Decomposition Lemma; 5.8 Notes and Comments; References; 6. Trees and the Art of Lower Bounds; 6.1 Introduction; 6.2 Trees; 6.3 A Toy Lower Bound; 6.4 Lower Bound for Theorem 4.3.2; 6.5 Lower Bound for Theorem 4.4.1; Reference; 7. Random Fourier Series and Trigonometric Sums, II; 7.1 Introduction; 7.2 Families of Distances; 7.3 Statement of Main Results; 7.4 Proofs, Lower Bounds; 7.5 Proofs, Upper Bounds; 7.6 Proofs, Convergence; 7.7 Explicit Computations; 7.8 Notes and Comments; References "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"--Publisher's description Based on recent developments, such as new isoperimetric inequalities and random process techniques, this book presents a thorough treatment of the main aspects of Probability in Banach spaces, and of some of their links to Geometry of Banach spaces. The fundamental question of characterizing continuity and boundedness of Gaussian processes goes back to Kolmogorov. This book provides an overview of "generic chaining", a completely natural variation on the ideas of Kolmogorov.