Markov Chains:
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cham
Springer
[2018]
|
| Schriftenreihe: | Springer series in operations research and financial engineering
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xviii, 757 Seiten Diagramme |
| ISBN: | 9783319977034 |
Internformat
MARC
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| 245 | 1 | 0 | |a Markov Chains |c Randal Douc ; Eric Moulines ; Pierre Priouret ; Philippe Soulier |
| 264 | 1 | |a Cham |b Springer |c [2018] | |
| 300 | |a xviii, 757 Seiten |b Diagramme | ||
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| 650 | 4 | |a Probability Theory and Stochastic Processes | |
| 650 | 4 | |a Distribution (Probability theory | |
| 650 | 0 | 7 | |a Markov-Kette |0 (DE-588)4037612-6 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Priouret, Pierre |d 1939- |0 (DE-588)173702538 |4 aut | |
| 700 | 1 | |a Soulier, Philippe |0 (DE-588)1178465055 |4 aut | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents Part I Foundations 1 Markov Chains: Basic Definitions........................................................... 1.1 Markov Chains................................................................................ 1.2 Kernels.............................................................................................. 1.3 Homogeneous Markov Chains...................................................... 1.4 Invariant Measures and Stationarity............................................... 1.5 Reversibility..................................................................................... 1.6 Markov Kernels on Lf(ī)................................................................ 1.7 Exercises............................................................................................ 1.8 Bibliographical Notes....................................................................... 3 3 6 12 16 18 20 21 25 2 Examples of Markov Chains.................................................................... 2.1 Random Iterative Functions........................................................... 2.2 Observation-Driven Models........................................................... 2.3 Markov Chain Monte Carlo Algorithms ...................................... 2.4 Exercises............................................................................................ 2.5 Bibliographical Notes....................................................................... 27 27 35 38 49 51 3 Stopping Times and the Strong Markov Property............................ 3.1 The Canonical
Chain....................................................................... 3.2 Stopping Times................................................................................ 3.3 The Strong Markov Property......................................................... 3.4 First-Entrance, Last-Exit Decomposition...................................... 3.5 Accessible and Attractive Sets...................................................... 3.6 Return Times and Invariant Measures.......................................... 3.7 Exercises............................................................................................ 3.8 Bibliographical Notes....................................................................... 53 54 58 60 64 66 67 73 74 xiii
4 5 XV Contents Martingales, Harmonic Functions and Poisson-Dirichlet Problems....................................................................................................... 4.1 Harmonic and Superharmonic Functions...................................... 4.2 The Potential Kemel....................................................................... 4.3 The Comparison Theorem............................................................. 4.4 The Dirichlet and Poisson Problems............................................. 4.5 Time-Inhomogeneous Poisson-Dirichlet Problems..................... 4.6 Exercises............................................................................................ 4.7 Bibliographical Notes....................................................................... Ergodic Theory for Markov Chains...................................................... 5.1 Dynamical Systems......................................................................... 5.2 Markov Chain Ergodicity................................................................ 5.3 Exercises............................................................................................ 5.4 Bibliographical Notes....................................................................... Contents 9 75 75 77 81 85 88 89 95 »jtøWtr*· xiv Part II 6 7 8 Irreducible Chains: Basics 11 Splitting Construction and Invariant Measures 11.1 The Splitting Construction............................................................. 11.2 Existence of Invariant Measures....................................................
11.3 Convergence in Total Variation tothe Stationary Distribution....................................................................................... 11.4 Geometric Convergence in Total Variation Distance ................. 11.5 Exercises............................................................................................ 11.6 Bibliographical Notes....................................................................... 1 l.A Another Proof of the Convergenceof Harris Recurrent Kemels.............................................................................................. 241 241 247 Feller and f-Kernete.................................................................................. 12.1 Feller Kernels............................................ 12.2 Г-Kemels ......................................................................................... 12.3 Existence of an Invariant Probability............................................. 12.4 Topological Recurrence.................................................................. 12.5 Exercises............................................................................................ 12.6 Bibliographical Notes....................................................................... 12.A Linear Control Systems................................................................... 265 265 270 274 277 279 285 285 ļ į ·. Atomic Chains............................................................................................ П9 6.1
Atoms................................................................................................. 119 6.2 Recurrence and Transience............................................................. 121 6.3 Period of an Atom........................................................................... 126 6.4 Subinvariant and Invariant Measures............................................. 128 6.5 Independence of the Excursions.................................................... 134 6.6 Ratio Limit Theorems .................................................................... 135 6.7 The Central Limit Theorem........................................................... 137 6.8 Exercises............................................................................................ 140 6.9 Bibliographical Notes....................................................................... 144 Markov Chains on aDiscrete StateSpace.............................................. 7.1 Irreducibility, Recurrence, and Transience................................... 7.2 Invariant Measures, Positive and Null Recurrence..................... 7.3 Communication................................................................................ 7.4 Period................................................................................................. 7.5 Drift Conditions for Recurrence and Transience.......................... 7.6 Convergence to the Invariant Probability...................................... 7.7 Exercises............................................................................................ 7.8
Bibliographical Notes....................................................................... 145 145 146 148 150 151 154 164 Convergence of Atomic MarkovChains................................................. 8.1 Discrete-Time Renewal Theory...................................................... 8.2 Renewal Theory and Atomic Markov Chains............................... 8.3 Coupling Inequalities for Atomic Markov Chains ..................... 8.4 Exercises............................................................................................ 8.5 Bibliographical Notes....................................................................... 165 165 175 180 187 189 159 211 215 215 Transience, Recurrence, and Harris Recurrence 221 10.1 Recurrence and Transience............................................................. 221 10.2 Harris Recurrence........................................................................... 228 10.3 Exercises............................................................................................ 236 10.4 Bibliographical Notes....................................................................... 239 •I įj 191 ^ 194 201 10 97 97 104 Ill 115 Small Sets, Irreducibility, and Aperiodicity 9.1 Small Sets......................................................................................... 9.2 Irreducibility..................................................................................... 9.3 Periodicity and Aperiodicity........................................................... 9.4 Petite
Sets......................................................................................... 9.5 Exercises............................................................................................ 9.6 Bibliographical Notes....................................................................... 9.A Proof of Theorem 9.2.6................................................................... 12 Part III 13 251 253 258 259 259 Irreducible Chains: Advanced Topics Rates 13.1 13.2 13.3 13.4 13.5 13.6 of Convergence for Atomic Markov Chains............................ Subgeometric Sequences.................................................................. Coupling Inequalities for Atomic Markov Chains ..................... Rates of Convergence in Total Variation Distance..................... Rates of Convergence in /-Norm.................................................. Exercises............................................................................................ Bibliographical Notes....................................................................... 289 289 291 303 305 311 312
XVI 14 15 16 17 18 19 Contents Geometric Recurrence and Regularity................................................. 14.1 /-Geometric Recurrence and Drift Conditions............................ 14.2 /-Geometric Regularity.................................................................... 14.3 /֊Geometric Regularity of the Skeletons...................................... 14.4 /-Geometric Regularity of the Split Kernel................................. 14.5 Exercises............................................................................................ 14.6 Bibliographical Notes....................................................................... 313 313 321 327 332 334 337 Geometric Rates of Convergence........................................................... 15.1 Geometric Ergodicity....................................................................... 15.2 V-Uniform Geometric Ergodicity.................................................... 15.3 Uniform Ergodicity......................................................................... 15.4 Exercises............................................................................................ 15.5 Bibliographical Notes....................................................................... 339 339 349 353 356 358 (ƒ, //Recurrence and Regularity........................................................... 361 16.1 (ƒ, /-/Recurrence and Drift Conditions........................................ 361 16.2 (ƒ, //Regularity.............................................................................. 370 16.3 (ƒ, //Regularity of
the Skeletons.................................................. 377 16.4 (ƒ, //Regularity of the Split Kernel............................................. 381 16.5 Exercises............................................................................................ 382 16.6 Bibliographical Notes....................................................................... 383 Subgeometric Rates of Convergence...................................................... 17.1 (ƒ, //Ergodicity.............................................................................. 17.2 Drift Conditions................................................................................ 17.3 Bibliographical Notes....................................................................... 17.A Young Functions............................................................................... 385 385 392 399 399 Uniform and V-Geometric Ergodicity by Operator Methods......... 18.1 The Fixed-Point Theorem................................................................ 18.2 Dobrushin Coefficient and Uniform Ergodicity............................ 18.3 V-Dobrushin Coefficient.................................................................. 18.4 V-Uniformly Geometrically Ergodic Markov Kernel................... 18.5 Application of Uniform Ergodicity to the Existence of an Invariant Measure.................................................................. 18.6 Exercises............................................................................................ 18.7 Bibliographical
Notes....................................................................... 401 401 403 409 412 Coupling for Irreducible Kernels........................................................... 19.1 Coupling............................................................................................ 19.2 The Coupling Inequality.................................................................. 19.3 Distributional, Exact, and Maximal Coupling............................... 19.4 A Coupling Proof of V-Geometric Ergodicity............................ 19.5 A Coupling Proof of Subgeometric Ergodicity............................ 421 422 432 435 441 444 Contents 19.6 19.7 Part IV 20 *vu Exercises............................................................................................ 449 Bibliographical Notes....................................................................... 451 Selected Topics Convergence in the Wasserstein Distance............................................. 20.1 The Wasserstein Distance................................................................ 20.2 Existence and Uniqueness of the Invariant Probability Measure............................................................................................ 20.3 Uniform Convergence in the Wasserstein Distance..................... 20.4 Nonuniform Geometric Convergence............................................. 20.5 Subgeometric Rates of Convergence for the Wasserstein Distance............................................................................................ 20.6
Exercises............................................................................................ 20.7 Bibliographical Notes....................................................................... 20.A Complements on the Wasserstein Distance.................................. 455 456 21 Central Limit Theorems........................................................................... 21.1 Preliminaries..................................................................................... 21.2 The Poisson Equation....................................................................... 21.3 The Resolvent Equation.................................................................. 21.4 A Martingale Coboundary Decomposition................................... 21.5 Exercises............................................................................................ 21.6 Bibliographical Notes....................................................................... 21.A A Covariance Inequality................................................................... 489 490 495 503 508 517 519 520 22 Spectral Theory.......................................................................................... 22.1 Spectrum............................................................................................ 22.2 Geometric and Exponential Convergence in L2(π)..................... 22.3 Lp(u)-Exponential Convergence.................................................... 22.4 Cheeger’s Inequality ..................... 22.5 Variance Bounds for Additive Functionals and the Central Limit Theorem for
Reversible Markov Chains............................ 22.6 Exercises............................................................................................ 22.7 Bibliographical Notes....................................................................... 22.A Operators on Banach and Hilbert Spaces....................................... 22.B Spectral Measure.............................................................................. 523 523 530 538 545 Concentration Inequalities ....................................................................... 23.1 Concentration Inequality for Independent Random Variables............................................................................................ 23.2 Concentration Inequality for Uniformly Ergodic Markov Chains................................................................................................. 23.3 Sub-Gaussian Concentration Inequalities for V-Geometrically Ergodic Markov Chains.................................................................. 575 415 417 419 23 462 465 471 476 480 485 486 553 560 562 563 572 576 581 587
xviii Contents 23.4 Exponential Concentration Inequalities Under Wasserstein Contraction....................................................................................... Exercises............................................................................................ Bibliographical Notes....................................................................... 594 599 601 Appendices............................................................................................................. 603 A Notations........................................................................................................ 605 В Topology, Measure and Probability....................................................... B.l Topology................................................................................................ B.2 Measures................................................................................................ B.3 Probability.............................................................................................. 609 609 612 618 C Weak Convergence....................................................................................... C. 1 Convergence on Locally Compact Metric Spaces........................... C.2 Tightness................................................................................................ 625 625 626 D Total and V-Total Variation Distances................................................... D.l Signed Measures.................................................................................. D.2 Total Variation
Distance..................................................................... D.3 V-Total Variation................................................................................ 629 629 631 635 E Martingales.................................................................................................... E.l Generalized Positive Supermartingales............................................. E.2 Martingales........................................................................................... E.3 Martingale Convergence Theorems................................................... E.4 Central Limit Theorems....................................................................... 637 637 638 639 641 F Mixing Coefficients...................................................................................... F.l Definitions.............................................................................................. F.2 Properties................................................................................................ F.3 Mixing Coefficients of Markov Chains............................................. 645 645 646 653 G Solutions to Selected Exercises................................................................ 657 References............................................................................................................. 733 Index...................................................................................................................... 753 23.5 23.6
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| author | Douc, Randal 1971- Moulines, Eric 1963- Priouret, Pierre 1939- Soulier, Philippe |
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| 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 |
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| id | DE-604.BV045443599 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T08:18:16Z |
| institution | BVB |
| isbn | 9783319977034 |
| language | English |
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| owner_facet | DE-188 DE-91G DE-BY-TUM DE-83 DE-739 |
| physical | xviii, 757 Seiten Diagramme |
| publishDate | 2018 |
| publishDateSearch | 2018 |
| publishDateSort | 2018 |
| publisher | Springer |
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| series2 | Springer series in operations research and financial engineering |
| spelling | Douc, Randal 1971- (DE-588)105018629X aut Markov Chains Randal Douc ; Eric Moulines ; Pierre Priouret ; Philippe Soulier Cham Springer [2018] xviii, 757 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Springer series in operations research and financial engineering Probability Theory and Stochastic Processes Distribution (Probability theory Markov-Kette (DE-588)4037612-6 gnd rswk-swf Markov-Kette (DE-588)4037612-6 s DE-604 Moulines, Eric 1963- (DE-588)1050741625 aut Priouret, Pierre 1939- (DE-588)173702538 aut Soulier, Philippe (DE-588)1178465055 aut Erscheint auch als Online-Ausgabe 978-3-319-97704-1 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=030829087&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Douc, Randal 1971- Moulines, Eric 1963- Priouret, Pierre 1939- Soulier, Philippe Markov Chains Probability Theory and Stochastic Processes Distribution (Probability theory Markov-Kette (DE-588)4037612-6 gnd |
| subject_GND | (DE-588)4037612-6 |
| title | Markov Chains |
| title_auth | Markov Chains |
| title_exact_search | Markov Chains |
| title_full | Markov Chains Randal Douc ; Eric Moulines ; Pierre Priouret ; Philippe Soulier |
| title_fullStr | Markov Chains Randal Douc ; Eric Moulines ; Pierre Priouret ; Philippe Soulier |
| title_full_unstemmed | Markov Chains Randal Douc ; Eric Moulines ; Pierre Priouret ; Philippe Soulier |
| title_short | Markov Chains |
| title_sort | markov chains |
| topic | Probability Theory and Stochastic Processes Distribution (Probability theory Markov-Kette (DE-588)4037612-6 gnd |
| topic_facet | Probability Theory and Stochastic Processes Distribution (Probability theory Markov-Kette |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030829087&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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