A weak convergence approach to the theory of large deviations:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Wiley
1997
|
| Schriftenreihe: | Wiley series in probability and statistics
A Wiley-interscience publication |
| Schlagworte: | |
| Online-Zugang: | Publisher description Table of Contents Inhaltsverzeichnis |
| Beschreibung: | XVII, 479 S. graph. Darst. |
| ISBN: | 0471076724 |
Internformat
MARC
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| 100 | 1 | |a Dupuis, Paul G. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a A weak convergence approach to the theory of large deviations |c Paul Dupuis ; Richard S. Ellis |
| 264 | 1 | |a New York [u.a.] |b Wiley |c 1997 | |
| 300 | |a XVII, 479 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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Datensatz im Suchindex
| _version_ | 1804126002593398784 |
|---|---|
| adam_text | Contents
1. Formulation of Large Deviation Theory in Terms of the Laplace
Principle 1
1.1. Introduction, 1
1.2. Equivalent Formulation of the Large Deviation
Principle, 4
1.3. Basic Results in the Theory, 16
1.4. Properties of the Relative Entropy, 32
1.5. Stochastic Control Theory and Dynamic Programming, 41
2. First Example: Sanov s Theorem 48
2.1. Introduction, 48
2.2. Statement of Sanov s Theorem, 49
2.3. The Representation Formula, 51
2.4. Proof of the Laplace Principle Lower Bound. 58
2.5. Proof of the Laplace Principle Upper Bound, 58
3. Second Example: Mogulskii s Theorem 65
3.1. Introduction, 65
3.2. The Representation Formula, 67
3.3. Proof of the Laplace Principle Upper Bound and
Identification of the Rate Function, 74
3.4. Statement of Mogulskii s Theorem and Completion of the
Proof, 81
3.5. Cramer s Theorem, 86
3.6. Comments on the Proofs, 88
xiii
Xiv CONTENTS
4. Representation Formulas for Other Stochastic Processes 92
4.1. Introduction, 92
4.2. The Representation Formula for the Empirical Measures
of a Markov Chain, 94
4.3. The Representation Formula for a Random Walk
Model, 98
4.4. The Representation Formula for a Random Walk Model
with State Dependent Noise, 102
4.5. Extensions to Unbounded Functions, 107
4.6. Representation Formulas for Continuous Time Markov
Processes, 111
4.6.1. Introduction, 111
4.6.2. Formal Derivation of Representation Formulas
for Continuous Time Markov Processes, 113
4.6.3. Examples of Continuous Time Representation
Formulas, 119
4.6.4. Remarks on the Proofs of the Representation
Formulas, 123
5. Compactness and Limit Properties for the Random
Walk Model 127
5.1. Introduction, 127
5.2. Definitions and a Representation Formula, 128
5.3. Compactness and Limit Properties, 131
5.4. Weaker Version of Condition 5.3.1, 147
6. Laplace Principle for the Random Walk Model with Continuous
Statistics 149
6.1. Introduction, 149
6.2. Proof of the Laplace Principle Upper Bound and
Identification of the Rate Function, 151
6.3. Statement of the Laplace Principle, 163
6.4. Strategy for the Proof of the Laplace Principle Lower
Bound, 171
6.5. Proof of the Laplace Principle Lower Bound Under
Conditions 6.2.1 and 6.3.1, 177
6.6. Proof of the Laplace Principle Lower Bound Under
Conditions 6.2.1 and 6.3.2, 185
6.7. Extension of Theorem 6.3.3 To Be Applied in
Chapter 10, 206
CONTENTS XV
7. Laplace Principle for the Random Walk Model with
Discontinuous Statistics 216
7.1. Introduction, 216
7.2. Statement of the Laplace Principle, 218
7.3. Laplace Principle for the Final Position Vectors and
One Dimensional Examples, 222
7.4. Proof of the Laplace Principle Upper Bound, 227
7.5. Proof of the Laplace Principle Lower Bound, 248
7.6. Compactness of the Level Sets of Ix, 270
8. Laplace Principle for the Empirical Measures of a Markov Chain 275
8.1. Introduction, 275
8.2. Compactness and Limit Properties of Controls and
Controlled Processes, 277
8.3. Proof of the Laplace Principle Upper Bound and
Identification of the Rate Function, 291
8.4. Statement of the Laplace Principle, 298
8.5. Properties of the Rate Function, 300
8.6. Proofs of the Laplace Principle Lower Bounds, 305
9. Extensions of the Laplace Principle for the Empirical Measures
of a Markov Chain 320
9.1. Introduction, 320
9.2. Laplace Principle for the Empirical Measures of a
Markov Chain with Discontinuous Statistics, 322
9.3. Laplace Limit for the Empirical Measures of a Markov
Chain in the t Topology, 332
10. Laplace Principle for Continuous Time Markov Processes with
Continuous Statistics 350
10.1. Introduction, 350
10.2. Statement of the Laplace Principle, 350
10.3. Proof of the Laplace Principle, 357
Appendix A. Background Material 371
A.I. Introduction, 371
A.2. Measure Theory, 371
A.3. Weak Convergence of Probability
Measures, 373
XVi CONTENTS
A.4. Probability Theory, 385
A.5. Stochastic Kernels, 389
A.6. Analysis 397
Appendix B. Deriving the Representation Formulas via Measure
Theory 400
B.I. Introduction, 400
B.2. Measure Theoretic Proof of the Representation
Formula for Sanov s Theorem, 400
B.3. Discussion, 403
Appendix C. Proofs of a Number of Results 405
C.I. Introduction, 405
C.2. Proof of the Donsker Varadhan Variational
Formula for the Relative Entropy, 405
C.3. Proof of the Chain Rule and Part (f) of
Lemma 1.4.3, 408
C.4. Proof of Part (g) of Lemma 1.4.3, 412
C.5. Proof of Part (f) of Lemma 6.2.3, 416
C.6. Proof of Part (g) of Lemma 6.2.3, 420
C.7. Proof of Proposition 6.3.4, 423
C.8. Continuity Property of Cramer Functions, 427
Appendix D. Convex Functions 431
D.I. Introduction, 431
D.2. Background Material on Convex
Functions, 431
D.3. Theorem on the Legendre Fenchel Transform
of Compositions of Convex Functions, 436
D.4. Three Examples, 445
Appendix E. Proof of Theorem 5.3.5 When Condition 5.4.1
Replaces Condition 5.3.1 449
E.I. Introduction, 449
E.2. Proofs of Results, 449
CONTENTS XVii
Bibliography 458
Notation Index 463
Author Index 467
Subject Index 469
|
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| discipline | Mathematik |
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| isbn | 0471076724 |
| language | English |
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| spelling | Dupuis, Paul G. Verfasser aut A weak convergence approach to the theory of large deviations Paul Dupuis ; Richard S. Ellis New York [u.a.] Wiley 1997 XVII, 479 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and statistics A Wiley-interscience publication Convergence (Mathématiques) ram Grandes déviations ram Large deviations Convergence Konvergenz (DE-588)4032326-2 gnd rswk-swf Große Abweichung (DE-588)4330658-5 gnd rswk-swf Schwache Konvergenz (DE-588)4180292-5 gnd rswk-swf Große Abweichung (DE-588)4330658-5 s Konvergenz (DE-588)4032326-2 s DE-604 Schwache Konvergenz (DE-588)4180292-5 s Ellis, Richard S. 1947-2018 Verfasser (DE-588)129738395 aut http://www.loc.gov/catdir/description/wiley034/96027513.html Publisher description http://www.loc.gov/catdir/toc/onix05/96027513.html Table of Contents HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007725716&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dupuis, Paul G. Ellis, Richard S. 1947-2018 A weak convergence approach to the theory of large deviations Convergence (Mathématiques) ram Grandes déviations ram Large deviations Convergence Konvergenz (DE-588)4032326-2 gnd Große Abweichung (DE-588)4330658-5 gnd Schwache Konvergenz (DE-588)4180292-5 gnd |
| subject_GND | (DE-588)4032326-2 (DE-588)4330658-5 (DE-588)4180292-5 |
| title | A weak convergence approach to the theory of large deviations |
| title_auth | A weak convergence approach to the theory of large deviations |
| title_exact_search | A weak convergence approach to the theory of large deviations |
| title_full | A weak convergence approach to the theory of large deviations Paul Dupuis ; Richard S. Ellis |
| title_fullStr | A weak convergence approach to the theory of large deviations Paul Dupuis ; Richard S. Ellis |
| title_full_unstemmed | A weak convergence approach to the theory of large deviations Paul Dupuis ; Richard S. Ellis |
| title_short | A weak convergence approach to the theory of large deviations |
| title_sort | a weak convergence approach to the theory of large deviations |
| topic | Convergence (Mathématiques) ram Grandes déviations ram Large deviations Convergence Konvergenz (DE-588)4032326-2 gnd Große Abweichung (DE-588)4330658-5 gnd Schwache Konvergenz (DE-588)4180292-5 gnd |
| topic_facet | Convergence (Mathématiques) Grandes déviations Large deviations Convergence Konvergenz Große Abweichung Schwache Konvergenz |
| url | http://www.loc.gov/catdir/description/wiley034/96027513.html http://www.loc.gov/catdir/toc/onix05/96027513.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007725716&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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