Upper and lower bounds for stochastic processes: decomposition theorems
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| Format: | Book |
| Language: | English |
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Cham
Springer
[2021]
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| Edition: | second edition |
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
Volume 60 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xviii, 726 Seiten Diagramme |
| ISBN: | 9783030825942 |
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| 245 | 1 | 0 | |a Upper and lower bounds for stochastic processes |b decomposition theorems |c Michel Talagrand |
| 250 | |a second edition | ||
| 264 | 1 | |a Cham |b Springer |c [2021] | |
| 300 | |a xviii, 726 Seiten |b Diagramme | ||
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| 490 | 1 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics |v Volume 60 | |
| 650 | 4 | |a Probability Theory and Stochastic Processes | |
| 650 | 4 | |a Functional Analysis | |
| 650 | 4 | |a Probabilities | |
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Contents 1 What Is This Book About?. 1.1 Philosophy. 1.2 What Is Chaining?. 1.3 The Kolmogorov Conditions. 1.4 Chaining in a Metric Space: Dudley’s Bound. 1.5 Overall Plan of the Book. 1.6 Does This Book Contain any Ideas?. 1.7 Overview by Chapters. 1.7.1 Gaussian Processes and the Generic Chaining. 1.7.2 Trees and Other Measures of Size. 1.7.3 Matching Theorems. 1.7.4 Warming Up with p-Stable Processes. 1.7.5 Bernoulli Processes. 1.7.6 Random Fourier Series and Trigonometric Sums . 1.7.7 Partition Scheme for Families of Distances. 1.7.8 Peaky Parts of Functions. 1.7.9 Proof of the Bernoulli Conjecture. 1.7.10 Random Series of Functions. 1.7.11 Infinitely Divisible Processes. 1.7.12 Unfulfilled Dreams.
1.7.13 Empirical Processes. 1.7.14 Gaussian Chaos. 1.7.15 Convergence of Orthogonal Series: Majorizing Measures. 1.7.16 Shor’s Matching Theorem. 1.7.17 The Ultimate Matching Theorem in Dimension Three. 1.7.18 Applications to Banach Space Theory. 1 1 2 2 6 8 9 10 10 11 12 12 12 13 14 14 14 15 15 15 15 16 16 16 17 17 xi
xii Contents Part I 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 3 3.2 3.3 3.4 3.5 4 Overview. Measuring the Size of the Supremum. The Union Bound and Other Basic Facts. The Generic Chaining. Entropy Numbers. Rolling Up Our Sleeves: Chaining in the Simplex. Admissible Sequences of Partitions. Functionals. Partitioning Schemes. Gaussian Processes: The Majorizing Measure Theorem. Gaussian Processes as Subsets of a Hilbert Space. Dreams.". A First Look at Ellipsoids. Rolling Up Our Sleeves: Chaining on Ellipsoids. Continuity of Gaussian Processes. Notes and Comments. Trees and Other Measures of Size. 3.1 Trees
. 3.1.1 Separated Trees. 3.1.2 Organized Trees. 3.1.3 Majorizing Measures. Rolling Up Our Sleeves: Trees in Ellipsoids . Fernique’s Functional. 3.3.1 Fernique’s Functional. 3.3.2 Fernique’s Convexity Argument. 3.3.3 From Majorizing Measures to Sequences of Partitions. Witnessing Measures. An Inequality of Fernique. Matching Theorems. 4.1 The Ellipsoid Theorem. 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 The Generic Chaining Gaussian Processes and the Generic Chaining. Partitioning Scheme II. Matchings. Discrepancy Bounds. The Ajtai-Komlós-Tusnády Matching Theorem. 4.5.1 The Long and
Instructive Way. 4.5.2 The Short and Magic Way. Lower Bound for the Ajtai-Komlós-Tusnády Theorem. The Leighton-Shor Grid Matching Theorem. Lower Bound for the Leighton-Shor Theorem. 21 21 21 23 28 32 38 40 45 49 59 63 71 73 78 81 84 87 87 88 90 92 95 98 98 100 102 104 106 Ill Ill 117 120 124 124 127 136 140 147 154 xiii Contents Part П 5 p-Stable Processes as Conditionally Gaussian Processes. A Lower Bound for p-Stable Processes. Philosophy. Simplification Through Abstraction . 1-Stable Processes. Where Do We Stand?. Bernoulli Processes. 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7 Some Dreams Come True Warming Up with p-Stable Processes 5.1 5.2 5.3 5.4 5.5 5.6 6 For the Expert Only. I58 Notes and Comments. 161 Bernoulli r.v.s. Boundedness of Bernoulli Processes. Concentration of
Measure. Sudakov Minoration. Comparison Principle. Control in Iой Norm. Peaky Parts of Functions . Discrepancy Bounds for Empirical Processes. Notes and Comments. Random Fourier Series and Trigonometric Sums . 7.1 7.2 7.3 7.4 7.5 Translation-Invariant Distances. Basics. 7.2.1 Simplification Through Abstraction. 7.2.2 Setting . 7.2.3 Upper Bounds in the Bernoulli Case. 7.2.4 Lower Bounds in the Gaussian Case. Random Distances. 7.3.1 Basic Principles. 7.3.2 A General Upper Bound. 7.3.3 A Side Story. The Marcus-Pisier Theorem. 7.4.1 The
Marcus-Pisier Theorem. 7.4.2 Applications of the Marcus-Pisier Theorem. Statement of Main Results . 7.5.1 General Setting. 7.5.2 Families of Distances. 7.5.3 Lower Bounds. 7.5.4 Upper Bounds. 7.5.5 Highlighting the Magic of Theorem 7.5.5 . 7.5.6 Combining Upper and Lower Bounds . 7.5.7 An Example: Tails in u~p . 165 165 167 167 169 172 173 175 175 177 180 182 187 189 192 197 200 201 203 2°6 206 207 208 209 2Ю 2Ю 213 215 218 218 220 223 223 225 227 230 230 232 232
xiv Contents 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 8 10 248 249 253 253 256 261 261 263 267 271 271 275 275 280 285 Tail Inequalities. 293 The Structure of Certain Canonical Processes. 300 Peaky Part of Functions. 309 9.1 9.2 9.3 9.4 9.5 Road Map. Peaky Part of Functions, II. Philosophy. Chaining for Bernoulli Processes. Notes and Comments. 309 309 319 320 323 Proof of the Bernoulli Conjecture. 325 10.1 Latala’s Principle. 326 10.2 10.3 10.4 10.5 10.6 Philosophy, I. Chopping Maps and Functionals. 10.3.1 Chopping Maps . 10.3.2 Basic Facts. 10.3.3 Functionals. Philosophy, II
. Latala’s Step. Philosophy, III. 10.7 A Decomposition Lemma. 10.8 Building the Partitions. 10.9 Philosophy, IV. 10.10 The Key Inequality. 10.11 Philosophy, V. 10.!2 Proof of the Latala-Bednorz Theorem. 10.13 Philosophy, VI . 10.14 A Geometric Characterization of b(T). 10.15 Lower Bounds from Measures. 10.16 Notes and Comments. 234 236 237 239 245 245 246 Partitioning Scheme and Families of Distances. 287 8.1 The Partitioning Scheme. 287 8.2 8.3 9 7.5.8 The Decomposition Theorem. 7.5.9 Convergence. A Primer on Random
Sets. Proofs, Lower Bounds. Proofs, Upper Bounds. 7.8.1 Road Map. 7.8.2 A Key Step. 7.8.3 Road Map: An Overview of Decomposition Theorems. 7.8.4 Decomposition Theorem in the Bernoulli Case. 7.8.5 Upper Bounds in the Bernoulli Case. 7.8.6 The Main Upper Bound. 7.8.7 Sums with Few Non-zero Terms. . . Proof of the Decomposition Theorem. 7.9.1 Constructing Decompositions. 7.9.2 Proof of Proposition 7.9.1 . Proofs, Convergence . Further Proofs. 7.11.1 Alternate Proof of Proposition 7.5.13. 7.11.2 Proof of Theorem 7.5.18. Explicit Computations. Vector-Valued Series: A Theoremof Fernique. Notes and
Comments. 331 331 331 336 340 343 344 347 xv Contents 11 348 351 355 356 362 363 366 367 370 372 Random Series of Functions. 373 11.1 11.2 11.3 11.4 11.5 11,6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 Road Map. Random Series of Functions: General Setting. Organization of the Chapter. The Main Lemma. Construction of the Majorizing Measure Using Convexity. From Majorizing Measures to Partitions. The General Lower Bound. The Giné-Zinn Inequalities. Proof of the Decomposition Theorem for Empirical Processes. The Decomposition Theorem for Random Series. Selector Processes and Why They Matter. Proving the Generalized Bernoulli Conjecture. Notes and Comments. 373 375 376 377 379 381 384 385 387 389 392 393 397 12 Infinitely Divisible Processes. 399 12.1 12.2 12.3 12.4 12.5 12.6
Poisson r.v.s and Poisson Point Processes. A Shortcut to Infinitely Divisible Processes. Overview of Results. 12.3.1 The Main Lower Bound . 12.3.2 The Decomposition Theorem for Infinitely Divisible Processes. 12.3.3 Upper Bounds ThroughBracketing. 12.3.4 Harmonizable InfinitelyDivisible Processes . 12.3.5 Example: Harmonizable p-Stable Processes . Proofs: The Bracketing Theorem. Proofs: The DecompositionTheorem for Infinitely Divisible Processes . Notes and Comments. 399 402 403 403 405 406 407 409 410 411 415 13 Unfulfilled Dreams . 417 13.1 13.2 Positive Selector Processes. 417 Explicitly Small Events. 418
Contents xvi 13.3 13.4 Part III 14 16 433 Bracketing. 433 The Class of Squares of a Given Class. 436 When Not to Use Chaining. 448 Gaussian Chaos. 15.1 Order 2 Gaussian Chaos. 15.1.1 Basic Facts. 15.1.2 WhenT Is Small for the Distance . 15.1.3 Covering Numbers.;. 15.1.4 Another Way to Bound S(T). 15.1.5 Yet Another Way to Bound S(T). 15.2 Tails of Multiple-Order Gaussian Chaos. 15.3 Notes and Comments. 19.3 19.4 A 493 495 497 501 506 510 520 529 545 547 549 Introduction. 549 The Discrepancy Theorem. 551 Lethal Weakness of the Approach. 557 The Ultimate Matching Theorem in Dimension 3 . 18.1 18.2 18.3 18.4 18.5 18.6 18.7 Introduction.
Regularization of φ . Discretization. Discrepancy Bound. Geometry. Probability, 1. Haar Basis Expansion. 561 561 565 568 569 573 580 587 19.1.1 Basic Definitions. 19.1.2 Operators from . 19.1.3 Computing the Cotype-2 Constant with Few Vectors . Unconditionality. 19.2.1 Classifying the Elements of Bi. 19.2.2 Subsets of Bi. 19.2.3 1-Unconditional Sequences and Gaussian Measures . Probabilistic Constructions. 19.3.1 Restriction of Operators. 19.3.2 The A(p)-Problem. Sidon Sets. Introduction. 653 Elements of Proof of the Upper
Bound. 654 The Lower Bound. 655 В Some Deterministic Arguments. B.l Hall’s Matching Theorem. B.2 Proof of Lemma 4.7.11. B.3 The Shor-Leighton Grid Matching Theorem. B.4 End of Proof of Theorem 17.2.1 . B.5 Proof of Proposition 17.3.1. B.6 Proof of Proposition 17.2.4. C Classical View of Infinitely Divisible Processes. C.l Infinitely Divisible Random Variables. C.2 C.3 C.4 D 605 605 605 607 609 618 618 621 626 630 631 640 644 Discrepancy for Convex Sets. 653 A.l A.2 A.3 Shor’s Matching Theorem . 17.1 17.2 17.3 18 457 457 457 462 468 470 471 474 492 Probability, II. 593 Final Effort. 596 Applications to Banach Space Theory . 19.1 Cotype of
Operators. 19.2 Convergence of Orthogonal Series: Majorizing Measures. 493 16.1 A Kind of Prologue: Chaining in a Metric Space and Pisier’s Bound. 16.2 Introduction to Orthogonal Series: Paszkiewicz’s Theorem. 16.3 Recovering the Classical Results. 16.4 Approach to Paszkiewicz’s Theorem: Bednorz’s Theorem. 16.5 Chaining, I. 16.6 Proof of Bednorz’s Theorem. 16.7 Permutations. 16.8 Chaining, II. 16.9 Chaining, III. 16.10 Notes and Comments. 17 19 Empirical Processes, II. 14.1 14.2 14.3 15 18.8 18.9 My Lifetime Favorite Problem. 421 Classes of Sets. 422 Practicing xvii Contents 677 677 Infinitely Divisible Processes. 679
Representation. 680 p-Stable Processes. 680 Reading Suggestions. D.l D.2 D.3 D.4 D.5 D.6 659 659 661 662 668 670 673 Partition Schemes. Geometry of Metric Spaces . Cover Times. Matchings. Super-concentration in the Sense of S. Chatterjee. High-Dimensional Statistics. 683 683 683 684 684 685 685
xviii Contents 687 687 687 688 688 E Research Directions. E. 1 The Latala-Bednorz Theorem. E.2 The Ultimate Matching Conjecture. E. 3 My Favorite Lifetime Problem. E.4 From a Set to Its Convex Hull. F Solutions of Selected Exercises. 689 G Comparison with the First Edition. 715 Bibliography. 717 Index. 725 |
| adam_txt |
Contents 1 What Is This Book About?. 1.1 Philosophy. 1.2 What Is Chaining?. 1.3 The Kolmogorov Conditions. 1.4 Chaining in a Metric Space: Dudley’s Bound. 1.5 Overall Plan of the Book. 1.6 Does This Book Contain any Ideas?. 1.7 Overview by Chapters. 1.7.1 Gaussian Processes and the Generic Chaining. 1.7.2 Trees and Other Measures of Size. 1.7.3 Matching Theorems. 1.7.4 Warming Up with p-Stable Processes. 1.7.5 Bernoulli Processes. 1.7.6 Random Fourier Series and Trigonometric Sums . 1.7.7 Partition Scheme for Families of Distances. 1.7.8 Peaky Parts of Functions. 1.7.9 Proof of the Bernoulli Conjecture. 1.7.10 Random Series of Functions. 1.7.11 Infinitely Divisible Processes. 1.7.12 Unfulfilled Dreams.
1.7.13 Empirical Processes. 1.7.14 Gaussian Chaos. 1.7.15 Convergence of Orthogonal Series: Majorizing Measures. 1.7.16 Shor’s Matching Theorem. 1.7.17 The Ultimate Matching Theorem in Dimension Three. 1.7.18 Applications to Banach Space Theory. 1 1 2 2 6 8 9 10 10 11 12 12 12 13 14 14 14 15 15 15 15 16 16 16 17 17 xi
xii Contents Part I 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 3 3.2 3.3 3.4 3.5 4 Overview. Measuring the Size of the Supremum. The Union Bound and Other Basic Facts. The Generic Chaining. Entropy Numbers. Rolling Up Our Sleeves: Chaining in the Simplex. Admissible Sequences of Partitions. Functionals. Partitioning Schemes. Gaussian Processes: The Majorizing Measure Theorem. Gaussian Processes as Subsets of a Hilbert Space. Dreams.". A First Look at Ellipsoids. Rolling Up Our Sleeves: Chaining on Ellipsoids. Continuity of Gaussian Processes. Notes and Comments. Trees and Other Measures of Size. 3.1 Trees
. 3.1.1 Separated Trees. 3.1.2 Organized Trees. 3.1.3 Majorizing Measures. Rolling Up Our Sleeves: Trees in Ellipsoids . Fernique’s Functional. 3.3.1 Fernique’s Functional. 3.3.2 Fernique’s Convexity Argument. 3.3.3 From Majorizing Measures to Sequences of Partitions. Witnessing Measures. An Inequality of Fernique. Matching Theorems. 4.1 The Ellipsoid Theorem. 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 The Generic Chaining Gaussian Processes and the Generic Chaining. Partitioning Scheme II. Matchings. Discrepancy Bounds. The Ajtai-Komlós-Tusnády Matching Theorem. 4.5.1 The Long and
Instructive Way. 4.5.2 The Short and Magic Way. Lower Bound for the Ajtai-Komlós-Tusnády Theorem. The Leighton-Shor Grid Matching Theorem. Lower Bound for the Leighton-Shor Theorem. 21 21 21 23 28 32 38 40 45 49 59 63 71 73 78 81 84 87 87 88 90 92 95 98 98 100 102 104 106 Ill Ill 117 120 124 124 127 136 140 147 154 xiii Contents Part П 5 p-Stable Processes as Conditionally Gaussian Processes. A Lower Bound for p-Stable Processes. Philosophy. Simplification Through Abstraction . 1-Stable Processes. Where Do We Stand?. Bernoulli Processes. 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7 Some Dreams Come True Warming Up with p-Stable Processes 5.1 5.2 5.3 5.4 5.5 5.6 6 For the Expert Only. I58 Notes and Comments. 161 Bernoulli r.v.s. Boundedness of Bernoulli Processes. Concentration of
Measure. Sudakov Minoration. Comparison Principle. Control in Iой Norm. Peaky Parts of Functions . Discrepancy Bounds for Empirical Processes. Notes and Comments. Random Fourier Series and Trigonometric Sums . 7.1 7.2 7.3 7.4 7.5 Translation-Invariant Distances. Basics. 7.2.1 Simplification Through Abstraction. 7.2.2 Setting . 7.2.3 Upper Bounds in the Bernoulli Case. 7.2.4 Lower Bounds in the Gaussian Case. Random Distances. 7.3.1 Basic Principles. 7.3.2 A General Upper Bound. 7.3.3 A Side Story. The Marcus-Pisier Theorem. 7.4.1 The
Marcus-Pisier Theorem. 7.4.2 Applications of the Marcus-Pisier Theorem. Statement of Main Results . 7.5.1 General Setting. 7.5.2 Families of Distances. 7.5.3 Lower Bounds. 7.5.4 Upper Bounds. 7.5.5 Highlighting the Magic of Theorem 7.5.5 . 7.5.6 Combining Upper and Lower Bounds . 7.5.7 An Example: Tails in u~p . 165 165 167 167 169 172 173 175 175 177 180 182 187 189 192 197 200 201 203 2°6 206 207 208 209 2Ю 2Ю 213 215 218 218 220 223 223 225 227 230 230 232 232
xiv Contents 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 8 10 248 249 253 253 256 261 261 263 267 271 271 275 275 280 285 Tail Inequalities. 293 The Structure of Certain Canonical Processes. 300 Peaky Part of Functions. 309 9.1 9.2 9.3 9.4 9.5 Road Map. Peaky Part of Functions, II. Philosophy. Chaining for Bernoulli Processes. Notes and Comments. 309 309 319 320 323 Proof of the Bernoulli Conjecture. 325 10.1 Latala’s Principle. 326 10.2 10.3 10.4 10.5 10.6 Philosophy, I. Chopping Maps and Functionals. 10.3.1 Chopping Maps . 10.3.2 Basic Facts. 10.3.3 Functionals. Philosophy, II
. Latala’s Step. Philosophy, III. 10.7 A Decomposition Lemma. 10.8 Building the Partitions. 10.9 Philosophy, IV. 10.10 The Key Inequality. 10.11 Philosophy, V. 10.!2 Proof of the Latala-Bednorz Theorem. 10.13 Philosophy, VI . 10.14 A Geometric Characterization of b(T). 10.15 Lower Bounds from Measures. 10.16 Notes and Comments. 234 236 237 239 245 245 246 Partitioning Scheme and Families of Distances. 287 8.1 The Partitioning Scheme. 287 8.2 8.3 9 7.5.8 The Decomposition Theorem. 7.5.9 Convergence. A Primer on Random
Sets. Proofs, Lower Bounds. Proofs, Upper Bounds. 7.8.1 Road Map. 7.8.2 A Key Step. 7.8.3 Road Map: An Overview of Decomposition Theorems. 7.8.4 Decomposition Theorem in the Bernoulli Case. 7.8.5 Upper Bounds in the Bernoulli Case. 7.8.6 The Main Upper Bound. 7.8.7 Sums with Few Non-zero Terms. . . Proof of the Decomposition Theorem. 7.9.1 Constructing Decompositions. 7.9.2 Proof of Proposition 7.9.1 . Proofs, Convergence . Further Proofs. 7.11.1 Alternate Proof of Proposition 7.5.13. 7.11.2 Proof of Theorem 7.5.18. Explicit Computations. Vector-Valued Series: A Theoremof Fernique. Notes and
Comments. 331 331 331 336 340 343 344 347 xv Contents 11 348 351 355 356 362 363 366 367 370 372 Random Series of Functions. 373 11.1 11.2 11.3 11.4 11.5 11,6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 Road Map. Random Series of Functions: General Setting. Organization of the Chapter. The Main Lemma. Construction of the Majorizing Measure Using Convexity. From Majorizing Measures to Partitions. The General Lower Bound. The Giné-Zinn Inequalities. Proof of the Decomposition Theorem for Empirical Processes. The Decomposition Theorem for Random Series. Selector Processes and Why They Matter. Proving the Generalized Bernoulli Conjecture. Notes and Comments. 373 375 376 377 379 381 384 385 387 389 392 393 397 12 Infinitely Divisible Processes. 399 12.1 12.2 12.3 12.4 12.5 12.6
Poisson r.v.s and Poisson Point Processes. A Shortcut to Infinitely Divisible Processes. Overview of Results. 12.3.1 The Main Lower Bound . 12.3.2 The Decomposition Theorem for Infinitely Divisible Processes. 12.3.3 Upper Bounds ThroughBracketing. 12.3.4 Harmonizable InfinitelyDivisible Processes . 12.3.5 Example: Harmonizable p-Stable Processes . Proofs: The Bracketing Theorem. Proofs: The DecompositionTheorem for Infinitely Divisible Processes . Notes and Comments. 399 402 403 403 405 406 407 409 410 411 415 13 Unfulfilled Dreams . 417 13.1 13.2 Positive Selector Processes. 417 Explicitly Small Events. 418
Contents xvi 13.3 13.4 Part III 14 16 433 Bracketing. 433 The Class of Squares of a Given Class. 436 When Not to Use Chaining. 448 Gaussian Chaos. 15.1 Order 2 Gaussian Chaos. 15.1.1 Basic Facts. 15.1.2 WhenT Is Small for the Distance . 15.1.3 Covering Numbers.;. 15.1.4 Another Way to Bound S(T). 15.1.5 Yet Another Way to Bound S(T). 15.2 Tails of Multiple-Order Gaussian Chaos. 15.3 Notes and Comments. 19.3 19.4 A 493 495 497 501 506 510 520 529 545 547 549 Introduction. 549 The Discrepancy Theorem. 551 Lethal Weakness of the Approach. 557 The Ultimate Matching Theorem in Dimension 3 . 18.1 18.2 18.3 18.4 18.5 18.6 18.7 Introduction.
Regularization of φ . Discretization. Discrepancy Bound. Geometry. Probability, 1. Haar Basis Expansion. 561 561 565 568 569 573 580 587 19.1.1 Basic Definitions. 19.1.2 Operators from . 19.1.3 Computing the Cotype-2 Constant with Few Vectors . Unconditionality. 19.2.1 Classifying the Elements of Bi. 19.2.2 Subsets of Bi. 19.2.3 1-Unconditional Sequences and Gaussian Measures . Probabilistic Constructions. 19.3.1 Restriction of Operators. 19.3.2 The A(p)-Problem. Sidon Sets. Introduction. 653 Elements of Proof of the Upper
Bound. 654 The Lower Bound. 655 В Some Deterministic Arguments. B.l Hall’s Matching Theorem. B.2 Proof of Lemma 4.7.11. B.3 The Shor-Leighton Grid Matching Theorem. B.4 End of Proof of Theorem 17.2.1 . B.5 Proof of Proposition 17.3.1. B.6 Proof of Proposition 17.2.4. C Classical View of Infinitely Divisible Processes. C.l Infinitely Divisible Random Variables. C.2 C.3 C.4 D 605 605 605 607 609 618 618 621 626 630 631 640 644 Discrepancy for Convex Sets. 653 A.l A.2 A.3 Shor’s Matching Theorem . 17.1 17.2 17.3 18 457 457 457 462 468 470 471 474 492 Probability, II. 593 Final Effort. 596 Applications to Banach Space Theory . 19.1 Cotype of
Operators. 19.2 Convergence of Orthogonal Series: Majorizing Measures. 493 16.1 A Kind of Prologue: Chaining in a Metric Space and Pisier’s Bound. 16.2 Introduction to Orthogonal Series: Paszkiewicz’s Theorem. 16.3 Recovering the Classical Results. 16.4 Approach to Paszkiewicz’s Theorem: Bednorz’s Theorem. 16.5 Chaining, I. 16.6 Proof of Bednorz’s Theorem. 16.7 Permutations. 16.8 Chaining, II. 16.9 Chaining, III. 16.10 Notes and Comments. 17 19 Empirical Processes, II. 14.1 14.2 14.3 15 18.8 18.9 My Lifetime Favorite Problem. 421 Classes of Sets. 422 Practicing xvii Contents 677 677 Infinitely Divisible Processes. 679
Representation. 680 p-Stable Processes. 680 Reading Suggestions. D.l D.2 D.3 D.4 D.5 D.6 659 659 661 662 668 670 673 Partition Schemes. Geometry of Metric Spaces . Cover Times. Matchings. Super-concentration in the Sense of S. Chatterjee. High-Dimensional Statistics. 683 683 683 684 684 685 685
xviii Contents 687 687 687 688 688 E Research Directions. E. 1 The Latala-Bednorz Theorem. E.2 The Ultimate Matching Conjecture. E. 3 My Favorite Lifetime Problem. E.4 From a Set to Its Convex Hull. F Solutions of Selected Exercises. 689 G Comparison with the First Edition. 715 Bibliography. 717 Index. 725 |
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| any_adam_object_boolean | 1 |
| author | Talagrand, Michel 1952- |
| author_GND | (DE-588)112924379 |
| author_facet | Talagrand, Michel 1952- |
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| author_sort | Talagrand, Michel 1952- |
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| bvnumber | BV047855007 |
| classification_rvk | SK 820 |
| ctrlnum | (OCoLC)1302308832 (DE-599)BVBBV047855007 |
| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | second edition |
| format | Book |
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| id | DE-604.BV047855007 |
| illustrated | Not Illustrated |
| index_date | 2024-07-03T19:15:54Z |
| indexdate | 2024-07-20T05:55:33Z |
| institution | BVB |
| isbn | 9783030825942 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-033237766 |
| oclc_num | 1302308832 |
| open_access_boolean | |
| owner | DE-83 DE-739 DE-824 |
| owner_facet | DE-83 DE-739 DE-824 |
| physical | xviii, 726 Seiten Diagramme |
| publishDate | 2021 |
| publishDateSearch | 2021 |
| publishDateSort | 2021 |
| publisher | Springer |
| record_format | marc |
| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics |
| spelling | Talagrand, Michel 1952- (DE-588)112924379 aut Upper and lower bounds for stochastic processes decomposition theorems Michel Talagrand second edition Cham Springer [2021] xviii, 726 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics Volume 60 Probability Theory and Stochastic Processes Functional Analysis Probabilities Functional analysis Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Untere Schranke (DE-588)4186970-9 gnd rswk-swf Obere Schranke (DE-588)4470960-2 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 s Obere Schranke (DE-588)4470960-2 s Untere Schranke (DE-588)4186970-9 s DE-604 Erscheint auch als Online-Ausgabe 978-3-030-82595-9 A Series of Modern Surveys in Mathematics Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge Volume 60 (DE-604)BV000899194 60 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033237766&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Talagrand, Michel 1952- Upper and lower bounds for stochastic processes decomposition theorems Probability Theory and Stochastic Processes Functional Analysis Probabilities Functional analysis Stochastischer Prozess (DE-588)4057630-9 gnd Untere Schranke (DE-588)4186970-9 gnd Obere Schranke (DE-588)4470960-2 gnd |
| subject_GND | (DE-588)4057630-9 (DE-588)4186970-9 (DE-588)4470960-2 |
| title | Upper and lower bounds for stochastic processes decomposition theorems |
| title_auth | Upper and lower bounds for stochastic processes decomposition theorems |
| title_exact_search | Upper and lower bounds for stochastic processes decomposition theorems |
| title_exact_search_txtP | Upper and lower bounds for stochastic processes decomposition theorems |
| title_full | Upper and lower bounds for stochastic processes decomposition theorems Michel Talagrand |
| title_fullStr | Upper and lower bounds for stochastic processes decomposition theorems Michel Talagrand |
| title_full_unstemmed | Upper and lower bounds for stochastic processes decomposition theorems Michel Talagrand |
| title_short | Upper and lower bounds for stochastic processes |
| title_sort | upper and lower bounds for stochastic processes decomposition theorems |
| title_sub | decomposition theorems |
| topic | Probability Theory and Stochastic Processes Functional Analysis Probabilities Functional analysis Stochastischer Prozess (DE-588)4057630-9 gnd Untere Schranke (DE-588)4186970-9 gnd Obere Schranke (DE-588)4470960-2 gnd |
| topic_facet | Probability Theory and Stochastic Processes Functional Analysis Probabilities Functional analysis Stochastischer Prozess Untere Schranke Obere Schranke |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033237766&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000899194 |
| work_keys_str_mv | AT talagrandmichel upperandlowerboundsforstochasticprocessesdecompositiontheorems |