Holdings: Probability and measure

Probability and measure:

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Bibliographic Details
Main Author:	Billingsley, Patrick 1925-2011 (Author)
Format:	Book
Language:	English
Published:	New York [u.a.] Wiley 1995
Edition:	3. ed.
Series:	Wiley series in probability and mathematical statistics
Subjects:	Meettechniek Mesure, Théorie de la Mesure, théorie de la Probabilités Waarschijnlijkheidstheorie Measure theory Probabilities Wahrscheinlichkeitstheorie Wahrscheinlichkeitsrechnung Maßtheorie
Online Access:	Inhaltsverzeichnis
Physical Description:	XII, 593 S. graph. Darst.
ISBN:	9780471007104 0471804789 0471007102

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Record in the Search Index

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adam_text	Contents CHAPTER 1. PROBABILITY 1. Borel s Normal Number Theorem, 1 The Unit Interval — The Weak Law of Large Numbers — The Strong Law of Large Numbers — Strong Law Versus Weak — Length — The Measure Theory of Diophantine Approximation* 2. Probability Measures, 17 Spaces — Assigning Probabilities — Classes of Sets — Probability Measures — Lebesgue Measure on the Unit Interval — Sequence Space* — Constructing σ -Fields* 3. Existence and Extension, 36 Construction of the Extension — Uniqueness and the π-λ Theorem — Monotone Classes — Lebesgue Measure on the Unit Interval — Completeness — Nonmeasurable Sets — Two Impossibility Theorems* 4. Denumerable Probabilities, 51 General Formulas — Limit Sets — Independent Events — Subfields — The Borel-Cantelli Lemmas — The Zero-One Law 5. Simple Random Variables, 67 Definition — Convergence of Random Variables — Independence — Existence of Independent Sequences — Expected Value — Inequalities 6. The Law of Large Numbers, 85 The Strong Law — The Weak Law — Bernstein s Theorem — A Refinement of the Second Borel-Cantelli Lemma Stars indicate topics that may be omitted on a first reading. vii VIU CONTENTS 7. Gambling Systems, 92 Gambler s Ruin — Selection Systems — Gambling Policies — Bold Play—Timid Play 8. Markov Chains, 111 Definitions — Higher-Order Transitions — An Existence Theorem — Transience and Persistence — Another Criterion for Persistence — Stationary Distributions — Exponential Convergence* — Optimal Stopping * 9. Large Deviations and the Law of the Iterated Logarithm,* 145 Moment Generating Functions — Large Deviations — Chernoff s Theorem — The Law of the Iterated Logarithm CHAPTER 2. MEASURE 158 10. General Measures, 158 Classes of Sets — Conventions Involving <»— Measures — Uniqueness 11. Outer Measure, 165 Outer Measure — Extension — An Approximation Theorem 12. Measures in Euclidean Space, 171 Lebesgue Measure — Regularity — Specifying Measures on the Line — Specifying Measures in Rk — Strange Euclidean Sets* 13. Measurable Functions and Mappings, 182 Measurable Mappings — Mappings into Rk — Limits and Measureability — Transformations of Measures 14. Distribution Functions, 187 Distribution Functions — Exponential Distributions — Weak Convergence — Convergence of Types* — Extremal Distributions* CHAPTER 3. INTEGRATION 199 15. The Integral, 199 Definition — Nonnegative Functions — Uniqueness CONTENTS 1X 16. Properties of the Integral, 206 Equalities and Inequalities — Integration to the Limit — Integration over Sets — Densities — Change of Variable — Uniform Integrability — Complex Functions 17. The Integral with Respect to Lebesgue Measure, 221 The Lebesgue Integral on the Line — The Riemann Integral — The Fundamental Theorem of Calculus — Change of Variable — The Lebesgue Integral in Rk — Stieltjes Integrals 18. Product Measure and Fubini s Theorem, 231 Product Spaces — Product Measure — Fubini s Theorem — Integration by Parts — Products of Higher Order 19. The L Spaces,* 241 Definitions — Completeness and Separability — Conjugate Spaces — Weak Compactness — Some Decision Theory — The Space L2 — An Estimation Problem CHAPTER 4. RANDOM VARIABLES AND EXPECTED VALUES 254 20. Random Variables and Distributions, 254 Random Variables and Vectors — Subfields — Distributions — Multidimensional Distributions — Independence — -Sequences of Random Variables — Convolution — Convergence in Probability — The Glivenko -Cantelli Theorem* 21. Expected Values, 273 Expected Value as Integral — Expected Values and Limits — Expected Values and Distributions — Moments — Inequalities — Joint Integrals — Independence and Expected Value — Moment Generating Functions 22. Sums of Independent Random Variables, 282 The Strong Law of Large Numbers — The Weak Law and Moment Generating Functions — Kolmogorov s Zero-One Law — Maximal Inequalities — Convergence of Random Series — Random Taylor Series * CONTENTS 23. The Poisson Process, 297 Characterization of the Exponential Distribution — The Poisson Process — The Poisson Approximation — Other Characterizations of the Poisson Process — Stochastic Processes 24. The Ergodic Theorem,* 310 Measure-Preserving Transformations — Ergodicity — Ergodicity of Rotations — Proof of the Ergodic Theorem — The Continued- Fraction Transformation — Diophantine Approximation CHAPTER 5. CONVERGENCE OF DISTRIBUTIONS 327 25. Weak Convergence, 327 Definitions — Uniform Distribution Modulo 1— Convergence in Distribution — Convergence in Probability — Fundamental Theorems — Helly s Theorem — Integration to the Limit 26. Characteristic Functions, 342 Definition — Moments and Derivatives — Independence—Inversion and the Uniqueness Theorem — The Continuity Theorem — Fourier Series 27. The Central Limit Theorem, 357 Identically Distributed Summands — The Lindeberg and Lyapounov Theorems — Dependent Variables 28. Infinitely Divisible Distributions,* 371 Vague Convergence — The Possible Limits — Characterizing the Limit 29. Limit Theorems in Rk, 378 The Basic Theorems — Characteristic Functions — Normal Distributions in Rk — The Central Limit Theorem 30. The Method of Moments,* 388 The Moment Problem — Moment Generating Functions — Central Limit Theorem by Moments — Application to Sampling Theory — Application to Number Theory CONTENTS Χ» CHAPTER 6. DERIVATIVES AND CONDITIONAL PROBABILITY 400 31. Derivatives on the Line,* 400 The Fundamental Theorem of Calculus — Derivatives of Integrals — Singular Functions — Integrals of Derivatives — Functions of Bounded Variation 32. The Radon-Nikodym Theorem, 419 Additive Set Functions — The Hahn Decomposition — Absolute Continuity and Singularity — The Main Theorem 33. Conditional Probability, 427 The Discrete Case — The General Case — Properties of Conditional Probability — Difficulties and Curiosities — Conditional Probability Distributions 34. Conditional Expectation, 445 Definition — Properties of Conditional Expectation — Conditional Distributions and Expectations — Sufficient Subfields* — Minimum-Variance Estimation * 35. Martingales, 458 Definition -Submartingdales — Gambling — Functions of Martingales — Stopping Times — Inequalities — Convergence Theorems — Applications: Derivatives — Likelihood Ratios — Reversed Martingales — Applications: de Finetti s Theorem — Bayes Estimation — A Central Limit Theorem* CHAPTER 7. STOCHASTIC PROCESSES 482 36. Kolmogorov s Existence Theorem, 482 Stochastic Processes — Finite-Dimensional Distributions — Product Spaces — Kolmogorov s Existence Theorem — The Inadequacy of â$T — A Return to Ergodic Theory* — The Hewitt-Savage Theorem * 37. Brownian Motion, 498 Definition — Continuity of Paths — Measurable Processes — Irregularity of Brownian Motion Paths — The Strong Markov Property — The Reflection Principle — Skorohod Embedding* — Invariance * Xli CONTENTS 38. Nondenumerable Probabilities,* 526 Introduction — Definitions — Existence Theorems—Consequences of Separability APPENDIX 536 NOTES ON THE PROBLEMS 552 BIBLIOGRAPHY 581 LIST OF SYMBOLS 585 INDEX 587
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author	Billingsley, Patrick 1925-2011
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dewey-tens	510 - Mathematics
discipline	Mathematik Wirtschaftswissenschaften
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spelling	Billingsley, Patrick 1925-2011 Verfasser (DE-588)171977858 aut Probability and measure Patrick Billingsley 3. ed. New York [u.a.] Wiley 1995 XII, 593 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and mathematical statistics Meettechniek gtt Mesure, Théorie de la Mesure, théorie de la ram Probabilités Probabilités ram Waarschijnlijkheidstheorie gtt Measure theory Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Maßtheorie (DE-588)4074626-4 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Maßtheorie (DE-588)4074626-4 s DE-188 Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s 1\p DE-604 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006866471&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk
spellingShingle	Billingsley, Patrick 1925-2011 Probability and measure Meettechniek gtt Mesure, Théorie de la Mesure, théorie de la ram Probabilités Probabilités ram Waarschijnlijkheidstheorie gtt Measure theory Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Maßtheorie (DE-588)4074626-4 gnd
subject_GND	(DE-588)4079013-7 (DE-588)4064324-4 (DE-588)4074626-4
title	Probability and measure
title_auth	Probability and measure
title_exact_search	Probability and measure
title_full	Probability and measure Patrick Billingsley
title_fullStr	Probability and measure Patrick Billingsley
title_full_unstemmed	Probability and measure Patrick Billingsley
title_short	Probability and measure
title_sort	probability and measure
topic	Meettechniek gtt Mesure, Théorie de la Mesure, théorie de la ram Probabilités Probabilités ram Waarschijnlijkheidstheorie gtt Measure theory Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Maßtheorie (DE-588)4074626-4 gnd
topic_facet	Meettechniek Mesure, Théorie de la Mesure, théorie de la Probabilités Waarschijnlijkheidstheorie Measure theory Probabilities Wahrscheinlichkeitstheorie Wahrscheinlichkeitsrechnung Maßtheorie
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