Probability: a survey of the mathematical theory
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Wiley
1996
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Wiley series in probability and statistics
A Wiley-interscience publication |
| Schlagworte: | |
| Online-Zugang: | Publisher description Table of Contents Inhaltsverzeichnis |
| Beschreibung: | IX, 189 S. |
| ISBN: | 0471154075 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804126002169774080 |
|---|---|
| adam_text | Contents
Preface to the Second Edition vii
Preface to the First Edition ix
Introduction 1
CHAPTER I
Elementary Probability Theory 5
§1. Probabilistic Model of an Experiment with a Finite Number of
Outcomes 5
§2. Some Classical Models and Distributions 17
§3. Conditional Probability. Independence 23
§4. Random Variables and Their Properties 32
§5. The Bernoulli Scheme. I. The Law of Large Numbers 45
§6. The Bernoulli Scheme. II. Limit Theorems (Local,
De Moivre Laplace, Poisson) 55
§7. Estimating the Probability of Success in the Bernoulli Scheme 70
§8. Conditional Probabilities and Mathematical Expectations with
Respect to Decompositions 76
§9. Random Walk. I. Probabilities of Ruin and Mean Duration in
Coin Tossing 83
§10. Random Walk. II. Reflection Principle. Arcsine Law 94
§11. Martingales. Some Applications to the Random Walk 103
§12. Markov Chains. Ergodic Theorem. Strong Markov Property 110
CHAPTER II
Mathematical Foundations of Probability Theory 131
§1. Probabilistic Model for an Experiment with Infinitely Many
Outcomes. Kolmogorov s Axioms 131
xiv Contents
§2. Algebras and tr algebras. Measurable Spaces 139
§3. Methods of Introducing Probability Measures on Measurable Spaces 151
§4. Random Variables. I. 170
§5. Random Elements 176
§6. Lebesgue Integral. Expectation 180
§7. Conditional Probabilities and Conditional Expectations with
Respect to a (T Algebra 212
§8. Random Variables. II. 234
§9. Construction of a Process with Given Finite Dimensional
Distribution 245
§10. Various Kinds of Convergence of Sequences of Random Variables 252
§11. The Hilbert Space of Random Variables with Finite Second Moment 262
§12. Characteristic Functions 274
§13. Gaussian Systems 297
CHAPTER III
Convergence of Probability Measures. Central Limit
Theorem 308
§1. Weak Convergence of Probability Measures and Distributions 308
§2. Relative Compactness and Tightness of Families of Probability
Distributions 317
§3. Proofs of Limit Theorems by the Method of Characteristic Functions 321
§4. Central Limit Theorem for Sums of Independent Random Variables. I.
The Lindeberg Condition 328
§5. Central Limit Theorem for Sums of Independent Random Variables. II.
Nonclassical Conditions 337
§6. Infinitely Divisible and Stable Distributions 341
§7. Metrizability of Weak Convergence 348
§8. On the Connection of Weak Convergence of Measures with Almost Sure
Convergence of Random Elements ( Method of a Single Probability
Space ) 353
§9. The Distance in Variation between Probability Measures.
Kakutani Hellinger Distance and Hellinger Integrals. Application to
Absolute Continuity and Singularity of Measures 359
§10. Contiguity and Entire Asymptotic Separation of Probability Measures 368
§11. Rapidity of Convergence in the Central Limit Theorem 373
§12. Rapidity of Convergence in Poisson s Theorem 376
CHAPTER IV
Sequences and Sums of Independent Random Variables 379
§1. Zero or One Laws 379
§2. Convergence of Series 384
§3. Strong Law of Large Numbers 388
§4. Law of the Iterated Logarithm 395
§5. Rapidity of Convergence in the Strong Law of Large Numbers and
in the Probabilities of Large Deviations 400
Contents xv
CHAPTER V
Stationary (Strict Sense) Random Sequences and
Ergodic Theory 404
§1. Stationary (Strict Sense) Random Sequences. Measure Preserving
Transformations 404
§2. Ergodicity and Mixing 407
§3. Ergodic Theorems 409
CHAPTER VI
Stationary (Wide Sense) Random Sequences. L2 Theory 415
§1. Spectral Representation of the Covariance Function 415
§2. Orthogonal Stochastic Measures and Stochastic Integrals 423
§3. Spectral Representation of Stationary (Wide Sense) Sequences 429
§4. Statistical Estimation of the Covariance Function and the Spectral
Density 440
§5. Wold s Expansion 446
§6. Extrapolation. Interpolation and Filtering 453
§7. The Kalman Bucy Filter and Its Generalizations 464
CHAPTER VII
Sequences of Random Variables that Form Martingales 474
§1. Definitions of Martingales and Related Concepts 474
§2. Preservation of the Martingale Property Under Time Change at a
Random Time 484
§3. Fundamental Inequalities 492
§4. General Theorems on the Convergence of Submartingales and
Martingales 508
§5. Sets of Convergence of Submartingales and Martingales 515
§6. Absolute Continuity and Singularity of Probability Distributions 524
§7. Asymptotics of the Probability of the Outcome of a Random Walk
with Curvilinear Boundary 536
§8. Central Limit Theorem for Sums of Dependent Random Variables 541
§9. Discrete Version of Ito s Formula 554
§10. Applications to Calculations of the Probability of Ruin in Insurance 558
CHAPTER VIII
Sequences of Random Variables that Form Markov Chains 564
§1. Definitions and Basic Properties 564
§2. Classification of the States of a Markov Chain in Terms of
Arithmetic Properties of the Transition Probabilities / ! 569
§3. Classification of the States of a Markov Chain in Terms of
Asymptotic Properties of the Probabilities pW 573
§4. On the Existence of Limits and of Stationary Distributions 582
§5. Examples 587
XVJ Contents
Historical and Bibliographical Notes 596
References 603
Index of Symbols 609
Index 611
|
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| spelling | Lamperti, John 1932- Verfasser (DE-588)132235676 aut Probability a survey of the mathematical theory John W. Lamperti 2. ed. New York [u.a.] Wiley 1996 IX, 189 S. txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and statistics A Wiley-interscience publication Probabilities Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Wahrscheinlichkeit (DE-588)4137007-7 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s DE-604 Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Wahrscheinlichkeit (DE-588)4137007-7 s 2\p DE-604 http://www.loc.gov/catdir/description/wiley032/96009559.html Publisher description http://www.loc.gov/catdir/toc/onix02/96009559.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=007725407&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 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lamperti, John 1932- Probability a survey of the mathematical theory Probabilities Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Wahrscheinlichkeit (DE-588)4137007-7 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4064324-4 (DE-588)4137007-7 (DE-588)4079013-7 (DE-588)4151278-9 |
| title | Probability a survey of the mathematical theory |
| title_auth | Probability a survey of the mathematical theory |
| title_exact_search | Probability a survey of the mathematical theory |
| title_full | Probability a survey of the mathematical theory John W. Lamperti |
| title_fullStr | Probability a survey of the mathematical theory John W. Lamperti |
| title_full_unstemmed | Probability a survey of the mathematical theory John W. Lamperti |
| title_short | Probability |
| title_sort | probability a survey of the mathematical theory |
| title_sub | a survey of the mathematical theory |
| topic | Probabilities Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Wahrscheinlichkeit (DE-588)4137007-7 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Probabilities Wahrscheinlichkeitsrechnung Wahrscheinlichkeit Wahrscheinlichkeitstheorie Einführung |
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