Probability with martingales:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge Univ. Press
1992
|
| Ausgabe: | Repr. |
| Schriftenreihe: | Cambridge mathematical textbooks
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 243 - 245 |
| Beschreibung: | XV, 251 S. |
| ISBN: | 0521406056 |
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Datensatz im Suchindex
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| adam_text | IMAGE 1
CONTENTS
PREFACE - PLEASE READ! XI
A QUESTION OF TERMINOLOGY XIII
A GUIDE TO NOTATION XIV
CHAPTER 0: A BRANCHING-PROCESS EXAMPLE 1
0.0. INTRODUCTORY REMARKS. 0.1. TYPICAL NUMBER OF CHILDREN, X. 0.2. SIZE
OF N TH GENERATION, Z N . 0.3. USE OF CONDITIONAL EXPECTATIONS. 0.4.
EXTINCTION PROBABILITY, TT. 0.5. PAUSE FOR THOUGHT: MEASURE. 0.6. OUR
FIRST MARTINGALE. 0.7. CONVERGENCE (OR NOT) OF EXPECTATIONS. 0.8.
FINDING THE DISTRIBUTION OF MOO- 0.9. CONCRETE EXAMPLE.
PART A: FOUNDATIONS
CHAPTER 1: MEASURE SPACES 14
1.0. INTRODUCTORY REMARKS. 1.1. DEFINITIONS OF ALGEBRA, CR-ALGEBRA. 1.2.
EX- AMPLES. BOREL ER-ALGEBRAS, B(S), B = B(R). 1.3. DEFINITIONS
CONCERNING SET FUNCTIONS. 1.4. DEFINITION OF MEASURE SPACE. 1.5.
DEFINITIONS CON- CERNING MEASURES. 1.6. LEMMA. UNIQUENESS OF EXTENSION,
7R-SYSTEMS. 1.7. THEOREM. CARATHEODORY S EXTENSION THEOREM. 1.8.
LEBESGUE MEASURE LEB ON ((0, L],#(0,1]). 1.9. LEMMA. ELEMENTARY
INEQUALITIES. 1.10. LEMMA.
MONOTONE-CONVERGENCE PROPERTIES OF MEASURES. 1.11. EXAMPLE/WARNING.
CHAPTER 2: EVENTS 23
2.1. MODEL FOR EXPERIMENT: (Q,,F,P). 2.2. THE INTUITIVE MEANING. 2.3.
EXAMPLES OF (FI,^ 7 ) PAIRS. 2.4. ALMOST SURELY (A.S.) 2.5. REMINDER:
LIMSUP,LIMINF,| LIM, ETC. 2.6. DEFINITIONS. LIMSUP.E N , (E N ,I.O.).
2.7.
IMAGE 2
FIRST BOREL-CANTELLI LEMMA (BC1). 2.8. DEFINITIONS, LIMINF E N ,(E N
,EV).
2.9. EXERCISE.
CHAPTER 3: RANDOM VARIABLES 29
3.1. DEFINITIONS. E-MEASURABLE FUNCTION, MS, (MS)+,BS. 3.2. ELEMENTARY
PROPOSITIONS ON MEASURABILITY. 3.3. LEMMA. SUMS AND PRODUCTS OF MEA-
SURABLE FUNCTIONS ARE MEASURABLE. 3.4. COMPOSITION LEMMA. 3.5. LEMMA ON
MEASURABILITY OF INFS, LIMINFS OF FUNCTIONS. 3.6. DEFINITION. RANDOM
VARIABLE. 3.7. EXAMPLE. COIN TOSSING. 3.8. DEFINITION, CR-ALGEBRA
GENERATED BY A COLLECTION OF FUNCTIONS ON FT. 3.9. DEFINITIONS. LAW,
DISTRIBUTION FUNC-
TION. 3.10. PROPERTIES OF DISTRIBUTION FUNCTIONS. 3.11. EXISTENCE OF
RANDOM VARIABLE WITH GIVEN DISTRIBUTION FUNCTION. 3.12. SKOROKOD
REPRESENTATION OF A RANDOM VARIABLE WITH PRESCRIBED DISTRIBUTION
FUNCTION. 3.13. GENERATED
CR-ALGEBRAS - A DISCUSSION. 3.14. THE MONOTONE-CLASS THEOREM.
CHAPTER 4: INDEPENDENCE 38
4.1. DEFINITIONS OF INDEPENDENCE. 4.2. THE JR-SYSTEM LEMMA; AND THE MORE
FAMILIAR DEFINITIONS. 4.3. SECOND BOREL-CANTELLI LEMMA (BC2). 4.4.
EXAMPLE. 4.5. A FUNDAMENTAL QUESTION FOR MODELLING. 4.6. A COIN-TOSSING
MODEL WITH APPLICATIONS. 4.7. NOTATION: IID RVS. 4.8. STOCHASTIC
PROCESSES;
MARKOV CHAINS. 4.9. MONKEY TYPING SHAKESPEARE. 4.10. DEFINITION. TAIL
CR- ALGEBRAS. 4.11. THEOREM. KOLMOGOROV S 0-1 LAW. 4.12.
EXERCISE/WARNING.
CHAPTER 5: INTEGRATION 49
5.0. NOTATION, ETC. /I(/) :=: F FD/I, /J,(F;A). 5.1. INTEGRALS OF
NON-NEGATIVE SIMPLE FUNCTIONS, SF + . 5.2. DEFINITION OF FI(F), F * (MS)
+ . 5.3. MONOTONE- CONVERGENCE THEOREM (MON). 5.4. THE FATOU LEMMAS FOR
FUNCTIONS (FA- TOU). 5.5. LINEARITY . 5.6. POSITIVE AND NEGATIVE PARTS
OF /. 5.7. INTE- GRABLE FUNCTION, C 1 (S, S,/I). 5.8. LINEARITY. 5.9.
DOMINATED CONVERGENCE
THEOREM (DOM). 5.10. SCHEFFE S LEMMA (SCHEFFE). 5.11. REMARK ON UNIFORM
INTEGRABILITY. 5.12. THE STANDARD MACHINE. 5.13. INTEGRALS OVER SUBSETS.
5.14. THE MEASURE FP, F G (M) + .
CHAPTER 6: EXPECTATION 58
INTRODUCTORY REMARKS. 6.1. DEFINITION OF EXPECTATION. 6.2. CONVERGENCE
THEOREMS. 6.3. THE NOTATION E(X;F). 6.4. MARKOV S INEQUALITY. 6.5. SUMS
OF NON-NEGATIVE RVS. 6.6. JENSEN S INEQUALITY FOR CONVEX FUNCTIONS. 6.7.
MONOTONICITY OF C P NORMS. 6.8. THE SCHWARZ INEQUALITY. 6.9. 2 :
PYTHAGORAS, COVARIANCE, ETC. 6.10. COMPLETENESS OF C (1 P OO). 6.11.
ORTHOGONAL PROJECTION. 6.12. THE ELEMENTARY FORMULA FOR EXPECTATION.
6.13. HOLDER FROM JENSEN.
IMAGE 3
CHAPTER 7: AN EASY STRONG LAW 71
7.1. INDEPENDENCE MEANS MULTIPLY - AGAIN! 7.2. STRONG LAW - FIRST
VERSION. 7.3. CHEBYSHEV S INEQUALITY. 7.4. WEIERSTRASS APPROXIMATION
THEOREM.
CHAPTER 8: PRODUCT MEASURE 75
8.0. INTRODUCTION AND ADVICE. 8.1. PRODUCT MEASURABLE STRUCTURE, SI X
S2. 8.2. PRODUCT MEASURE, FUBINI S THEOREM. 8.3. JOINT LAWS, JOINT PDFS.
8.4. INDEPENDENCE AND PRODUCT MEASURE. 8.5. B(R) N = B(R N ). 8.6. THE
N-FOLD EXTENSION. 8.7. INFINITE PRODUCTS OF PROBABILITY TRIPLES. 8.8.
TECHNICAL NOTE ON THE EXISTENCE OF JOINT LAWS.
PART B: MARTINGALE THEORY
CHAPTER 9: CONDITIONAL EXPECTATION 83
9.1. A MOTIVATING EXAMPLE. 9.2. FUNDAMENTAL THEOREM AND DEFINITION
(KOLMOGOROV, 1933). 9.3. THE INTUITIVE MEANING. 9.4. CONDITIONAL EX-
PECTATION AS LEAST-SQUARES-BEST PREDICTOR. 9.5. PROOF OF THEOREM 9.2.
9.6. AGREEMENT WITH TRADITIONAL EXPRESSION. 9.7. PROPERTIES OF
CONDITIONAL EX- PECTATION: A LIST. 9.8. PROOFS OF THE PROPERTIES IN
SECTION 9.7. 9.9. REGULAR
CONDITIONAL PROBABILITIES AND PDFS. 9.10. CONDITIONING UNDER
INDEPENDENCE ASSUMPTIONS. 9.11. USE OF SYMMETRY: AN EXAMPLE.
CHAPTER 10: MARTINGALES 93
10.1. FILTERED SPACES. 10.2. ADAPTED PROCESSES. 10.3. MARTINGALE, SUPER-
MARTINGALE, SUBMARTINGALE. 10.4. SOME EXAMPLES OF MARTINGALES. 10.5.
FAIR AND UNFAIR GAMES. 10.6. PREVISIBLE PROCESS, GAMBLING STRATEGY.
10.7. A FUN- DAMENTAL PRINCIPLE: YOU CAN T BEAT THE SYSTEM! 10.8.
STOPPING TIME. 10.9.
STOPPED SUPERMARTINGALES ARE SUPERMARTINGALES. 10.10. DOOB S OPTIONAL-
STOPPING THEOREM. 10.11. AWAITING THE ALMOST INEVITABLE. 10.12. HITTING
TIMES FOR SIMPLE RANDOM WALK. 10.13. NON-NEGATIVE SUPERHARMONIC FUNC-
TIONS FOR MARKOV CHAINS.
CHAPTER 11: THE CONVERGENCE THEOREM 106
11.1. THE PICTURE THAT SAYS IT ALL. 11.2. UPCROSSINGS. 11.3. DOOB S
UPCROSS- ING LEMMA. 11.4. COROLLARY. 11.5. DOOB S FORWARD CONVERGENCE
THEOREM. 11.6. WARNING. 11.7. COROLLARY.
IMAGE 4
CHAPTER 12: MARTINGALES BOUNDED IN C 2 110
12.0. INTRODUCTION. 12.1. MARTINGALES IN C 2 : ORTHOGONALITY OF
INCREMENTS. 12.2. SUMS OF ZERO-MEAN INDEPENDENT RANDOM VARIABLES IN C 2
. 12.3. RAN- DOM SIGNS. 12.4. A SYMMETRIZATION TECHNIQUE: EXPANDING THE
SAMPLE SPACE. 12.5. KOLMOGOROV S THREE-SERIES THEOREM. 12.6. CESARO S
LEMMA. 12.7.
KRONECKER S LEMMA. 12.8. A STRONG LAW UNDER VARIANCE CONSTRAINTS. 12.9.
KOLMOGOROV S TRUNCATION LEMMA. 12.10. KOLMOGOROV S STRONG LAW OF LARGE
NUMBERS (SLLN). 12.11. DOOB DECOMPOSITION. 12.12. THE ANGLE- BRACKETS
PROCESS (M). 12.13. RELATING CONVERGENCE OF M TO FINITENESS OF
(M)OC,. 12.14. A TRIVIAL STRONG LAW FOR MARTINGALES IN C?. 12.15.
LEVY S EXTENSION OF THE BOREL-CANTELLI LEMMAS. 12.16. COMMENTS.
CHAPTER 13: UNIFORM INTEGRABILITY 126
13.1. AN ABSOLUTE CONTINUITY PROPERTY. 13.2. DEFINITION. UI FAMILY.
13.3. TWO SIMPLE SUFFICIENT CONDITIONS FOR THE UI PROPERTY. 13.4. UI
PROPERTY OF CONDITIONAL EXPECTATIONS. 13.5. CONVERGENCE IN PROBABILITY.
13.6. ELE- MENTARY PROOF OF (BDD). 13.7. A NECESSARY AND SUFFICIENT
CONDITION FOR C 1
CONVERGENCE.
CHAPTER 14: UI MARTINGALES 133
14.0. INTRODUCTION. 14.1. UI MARTINGALES. 14.2. LEVY S UPWARD THEOREM.
14.3. MARTINGALE PROOF OF KOLMOGOROV S 0-1 LAW. 14.4. LEVY S DOWNWARD
THEOREM. 14.5. MARTINGALE PROOF OF THE STRONG LAW. 14.6. DOOB S SUB-
MARTINGALE INEQUALITY. 14.7. LAW OF THE ITERATED LOGARITHM: SPECIAL
CASE. 14.8. A STANDARD ESTIMATE ON THE NORMAL DISTRIBUTION. 14.9.
REMARKS ON EX- PONENTIAL BOUNDS; LARGE DEVIATION THEORY. 14.10. A
CONSEQUENCE OF HOLDER S INEQUALITY. 14.11. DOOB S C P INEQUALITY. 14.12.
KAKUTANI S THEOREM ON
PRODUCT MARTINGALES. 14.13.THE RADON-NIKODYM THEOREM. 14.14. THE
RADON-NIKODYM THEOREM AND CONDITIONAL EXPECTATION. 14.15. LIKELIHOOD
RATIO; EQUIVALENT MEASURES. 14.16. LIKELIHOOD RATIO AND CONDITIONAL
EXPEC- TATION. 14.17. KAKUTANI S THEOREM REVISITED; CONSISTENCY OF LR
TEST. 14.18. NOTE ON HARDY SPACES, ETC.
CHAPTER 15: APPLICATIONS 153
15.0. INTRODUCTION - PLEASE READ! 15.1. A TRIVIAL
MARTINGALE-REPRESENTATION RESULT. 15.2. OPTION PRICING; DISCRETE
BLACK-SCHOLES FORMULA. 15.3. THE MABINOGION SHEEP PROBLEM. 15.4. PROOF
OF LEMMA 15.3(C). 15.5. PROOF OF RESULT 15.3(D). 15.6. RECURSIVE NATURE
OF CONDITIONAL PROBABILITIES. 15.7.
BAYES FORMULA FOR BIVARIATE NORMAL DISTRIBUTIONS. 15.8. NOISY
OBSERVATION OF A SINGLE RANDOM VARIABLE. 15.9. THE KALMAN-BUCY FILTER.
15.10. HARNESSES ENTANGLED. 15.11. HARNESSES UNRAVELLED, 1. 15.12.
HARNESSES UNRAVELLED, 2.
IMAGE 5
PART C: CHARACTERISTIC FUNCTIONS
CHAPTER 16: BASIC PROPERTIES OF CFS 172
16.1. DEFINITION. 16.2. ELEMENTARY PROPERTIES. 16.3. SOME USES OF CHAR-
ACTERISTIC FUNCTIONS. 16.4. THREE KEY RESULTS. 16.5. ATOMS. 16.6. LEVY S
INVERSION FORMULA. 16.7. A TABLE.
CHAPTER 17: WEAK CONVERGENCE 179
17.1. THE ELEGANT DEFINITION. 17.2. A PRACTICAL FORMULATION. 17.3.
SKO- ROKHOD REPRESENTATION. 17.4. SEQUENTIAL COMPACTNESS FOR PROB(R).
17.5. TIGHTNESS.
CHAPTER 18: THE CENTRAL LIMIT THEOREM 185
18.1. LEVY S CONVERGENCE THEOREM. 18.2. O AND 0 NOTATION. 18.3. SOME
IMPORTANT ESTIMATES. 18.4. THE CENTRAL LIMIT THEOREM. 18.5. EXAMPLE.
18.6. CF PROOF OF LEMMA 12.4.
APPENDICES
CHAPTER A L: APPENDIX TO CHAPTER 1 192
AL.L. A NON-MEASURABLE SUBSET A OF S 1 . AL.2. (/-SYSTEMS. A1.3.
DYNKIN S LEMMA. A1.4. PROOF OF UNIQUENESS LEMMA 1.6. AL.5. A-SETS:
ALGEBRA CASE. A1.6. OUTER MEASURES. AL.7. CARATHEODORY S LEMMA. A1.8.
PROOF OF CARATHEODORY S THEOREM. A1.9. PROOF OF THE EXISTENCE OF
LEBESGUE MEASURE
ON ((0, L],B(0, 1]). A1.10. EXAMPLE OF NON-UNIQUENESS OF EXTENSION.
AL.LL. COMPLETION OF A MEASURE SPACE. AL.12. THE BAIRE CATEGORY THEOREM.
CHAPTER A3: APPENDIX TO CHAPTER 3 205
A3.1. PROOF OF THE MONOTONE-CLASS THEOREM 3.14. A3.2. DISCUSSION OF
GENERATED CR-ALGEBRAS.
CHAPTER A4: APPENDIX TO CHAPTER 4 208
A4.1. KOLMOGOROV S LAW OF THE ITERATED LOGARITHM. A4.2. STRASSEN S LAW
OF THE ITERATED LOGARITHM. A4.3. A MODEL FOR A MARKOV CHAIN.
CHAPTER A5: APPENDIX TO CHAPTER 5 211
A5.1. DOUBLY MONOTONE ARRAYS. A5.2. THE KEY USE OF LEMMA 1.10(A). A5.3.
UNIQUENESS OF INTEGRAL . A5.4. PROOF OF THE MONOTONE-CONVERGENCE
THEOREM.
IMAGE 6
CHAPTER A9: APPENDIX TO CHAPTER 9 214
A9.1. INFINITE PRODUCTS: SETTING THINGS UP. A9.2. PROOF OF A9.1(E).
CHAPTER A13: APPENDIX TO CHAPTER 13 217
A13.1. MODES OF CONVERGENCE: DEFINITIONS. A13.2. MODES OF CONVERGENCE:
RELATIONSHIPS.
CHAPTER A14: APPENDIX TO CHAPTER 14 219
A14.1. THE CR-ALGEBRA TT, T A STOPPING TIME. A14.2. A SPECIAL CASE OF
OST. A14.3. DOOB S OPTIONAL-SAMPLING THEOREM FOR UI MARTINGALES. A14.4.
THE RESULT FOR UI SUBMARTINGALES.
CHAPTER A16: APPENDIX TO CHAPTER 16 222
A16.1. DIFFERENTIATION UNDER THE INTEGRAL SIGN.
CHAPTER E: EXERCISES 224
REFERENCES 243
INDEX 246
|
| any_adam_object | 1 |
| author | Williams, David |
| author_facet | Williams, David |
| author_role | aut |
| author_sort | Williams, David |
| author_variant | d w dw |
| building | Verbundindex |
| bvnumber | BV009534410 |
| classification_rvk | SK 800 SK 820 |
| classification_tum | MAT 605f |
| ctrlnum | (OCoLC)257868085 (DE-599)BVBBV009534410 |
| dewey-full | 519.287 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.287 |
| dewey-search | 519.287 |
| dewey-sort | 3519.287 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Repr. |
| format | Book |
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| id | DE-604.BV009534410 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:36:40Z |
| institution | BVB |
| isbn | 0521406056 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006296142 |
| oclc_num | 257868085 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM |
| owner_facet | DE-91G DE-BY-TUM |
| physical | XV, 251 S. |
| publishDate | 1992 |
| publishDateSearch | 1992 |
| publishDateSort | 1992 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series2 | Cambridge mathematical textbooks |
| spelling | Williams, David Verfasser aut Probability with martingales Repr. Cambridge Cambridge Univ. Press 1992 XV, 251 S. txt rdacontent n rdamedia nc rdacarrier Cambridge mathematical textbooks Literaturverz. S. 243 - 245 Martingale - (Wahrscheinlichkeitsrechnung) idsbb Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Martingal (DE-588)4126466-6 gnd rswk-swf Martingaltheorie (DE-588)4168982-3 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Martingal (DE-588)4126466-6 s DE-188 Martingaltheorie (DE-588)4168982-3 s Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s 1\p DE-604 2\p DE-604 SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006296142&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Williams, David Probability with martingales Martingale - (Wahrscheinlichkeitsrechnung) idsbb Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Martingal (DE-588)4126466-6 gnd Martingaltheorie (DE-588)4168982-3 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4064324-4 (DE-588)4126466-6 (DE-588)4168982-3 (DE-588)4079013-7 |
| title | Probability with martingales |
| title_auth | Probability with martingales |
| title_exact_search | Probability with martingales |
| title_full | Probability with martingales |
| title_fullStr | Probability with martingales |
| title_full_unstemmed | Probability with martingales |
| title_short | Probability with martingales |
| title_sort | probability with martingales |
| topic | Martingale - (Wahrscheinlichkeitsrechnung) idsbb Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Martingal (DE-588)4126466-6 gnd Martingaltheorie (DE-588)4168982-3 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Martingale - (Wahrscheinlichkeitsrechnung) Wahrscheinlichkeitsrechnung Martingal Martingaltheorie Wahrscheinlichkeitstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006296142&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT williamsdavid probabilitywithmartingales |