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Produktbild: Probability and Measure

Probability and Measure Anniversary Edition

172,99 €

inkl. gesetzl. MwSt., Versandkostenfrei

Lieferung nach Hause

Beschreibung

Produktdetails

Einband

Gebundene Ausgabe

Erscheinungsdatum

09.03.2012

Verlag

John Wiley & Sons Inc

Seitenzahl

688

Maße (L/B/H)

26/18,3/3,9 cm

Gewicht

1390 g

Auflage

4. Auflage

Sprache

Englisch

ISBN

978-1-118-12237-2

Beschreibung

Rezension

"Like the previous editions, this Anniversary edition will be well received by students of mathematics, statistics, economics, and a wide variety of disciplines that require a solid understanding of probability theory." ( Int. J. Microstructure and Materials Properties , 1 February 2013)

Produktdetails

Einband

Gebundene Ausgabe

Erscheinungsdatum

09.03.2012

Verlag

John Wiley & Sons Inc

Seitenzahl

688

Maße (L/B/H)

26/18,3/3,9 cm

Gewicht

1390 g

Auflage

4. Auflage

Sprache

Englisch

ISBN

978-1-118-12237-2

Herstelleradresse

Libri GmbH
Europaallee 1
36244 Bad Hersfeld
DE

Email: gpsr@libri.de

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  • Produktbild: Probability and Measure
  • FOREWORD xi

    PREFACE xiii

    Patrick Billingsley 1925-2011 xv

    Chapter1 PROBABILITY 1

    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, 18

    Spaces

    Assigning Probabilities

    Classes of Sets

    Probability Measures

    Lebesgue Measure on the Unit Interval

    Sequence Space*

    Constructing s-Fields*

    3. EXISTENCE AND EXTENSION, 39

    Construction of the Extension

    Uniqueness and the p-? Theorem

    Monotone Classes

    Lebesgue Measure on the Unit Interval

    Completeness

    Nonmeasurable Sets

    Two Impossibility Theorems*

    4. DENUMERABLE PROBABILITIES, 53

    General Formulas

    Limit Sets

    Independent Events

    Subfields

    The Borel-Cantelli Lemmas

    The Zero-One Law

    5. SIMPLE RANDOM VARIABLES, 72

    Definition

    Convergence of Random Variables

    Independence

    Existence of Independent Sequences

    Expected Value

    Inequalities

    6. THE LAW OF LARGE NUMBERS, 90

    The Strong Law

    The Weak Law

    Bernstein's Theorem

    A Refinement of the Second Borel-Cantelli Lemma

    7. GAMBLING SYSTEMS, 98

    Gambler's Ruin

    Selection Systems

    Gambling Policies

    Bold Play*

    Timid Play*

    8. MARKOV CHAINS, 117

    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, 154

    Moment Generating Functions

    Large Deviations

    Chernoff's Theorem*

    The Law of the Iterated Logarithm

    Chapter2 MEASURE 167

    10. GENERAL MEASURES, 167

    Classes of Sets

    Conventions Involving 8

    Measures

    Uniqueness

    11. OUTER MEASURE, 174

    Outer Measure

    Extension

    An Approximation Theorem

    12. MEASURES IN EUCLIDEAN SPACE, 181

    Lebesgue Measure

    Regularity

    Specifying Measures on the Line

    Specifying Measures in Rk

    Strange Euclidean Sets*

    13. MEASURABLE FUNCTIONS AND MAPPINGS, 192

    Measurable Mappings

    Mappings into Rk

    Limits and Measurability

    Transformations of Measures

    14. DISTRIBUTION FUNCTIONS, 198

    Distribution Functions

    Exponential Distributions

    Weak Convergence

    Convergence of Types*

    Extremal Distributions*

    Chapter3 INTEGRATION 211

    15. THE INTEGRAL, 211

    Definition

    Nonnegative Functions

    Uniqueness

    16. PROPERTIES OF THE INTEGRAL, 218

    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, 234

    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, 245

    Product Spaces

    Product Measure

    Fubini's Theorem

    Integration by Parts

    Products of Higher Order

    19. THE Lp SPACES*, 256

    Definitions

    Completeness and Separability

    Conjugate Spaces

    Weak Compactness

    Some Decision Theory

    The Space L2

    An Estimation Problem

    Chapter4 RANDOM VARIABLES AND EXPECTED VALUES 271

    20. RANDOM VARIABLES AND DISTRIBUTIONS, 271

    Random Variables and Vectors

    Subfields

    Distributions

    Multidimensional Distributions

    Independence

    Sequences of Random Variables

    Convolution

    Convergence in Probability

    The Glivenko-Cantelli Theorem*

    21. EXPECTED VALUES, 291

    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, 300

    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*

    23. THE POISSON PROCESS, 316

    Characterization of the Exponential Distribution

    The Poisson Process

    The Poisson Approximation

    Other Characterizations of the Poisson Process

    Stochastic Processes

    24. THE ERGODIC THEOREM*, 330

    Measure-Preserving Transformations

    Ergodicity

    Ergodicity of Rotations

    Proof of the Ergodic Theorem

    The Continued-Fraction Transformation

    Diophantine Approximation

    Chapter5 CONVERGENCE OF DISTRIBUTIONS 349

    25. WEAK CONVERGENCE, 349

    Definitions

    Uniform Distribution Modulo 1*

    Convergence in Distribution

    Convergence in Probability

    Fundamental Theorems

    Helly's Theorem

    Integration to the Limit

    26. CHARACTERISTIC FUNCTIONS, 365

    Definition

    Moments and Derivatives

    Independence

    Inversion and the Uniqueness Theorem

    The Continuity Theorem

    Fourier Series*

    27. THE CENTRAL LIMIT THEOREM, 380

    Identically Distributed Summands

    The Lindeberg and Lyapounov Theorems

    Dependent Variables*

    28. INFINITELY DIVISIBLE DISTRIBUTIONS*, 394

    Vague Convergence

    The Possible Limits

    Characterizing the Limit

    29. LIMIT THEOREMS IN Rk, 402

    The Basic Theorems

    Characteristic Functions

    Normal Distributions in Rk

    The Central Limit Theorem

    30. THE METHOD OF MOMENTS*, 412

    The Moment Problem

    Moment Generating Functions

    Central Limit Theorem by Moments

    Application to Sampling Theory

    Application to Number Theory

    Chapter6 DERIVATIVES AND CONDITIONAL PROBABILITY 425

    31. DERIVATIVES ON THE LINE*, 425

    The Fundamental Theorem of Calculus

    Derivatives of Integrals

    Singular Functions

    Integrals of Derivatives

    Functions of Bounded Variation

    32. THE RADON-NIKODYM THEOREM, 446

    Additive Set Functions

    The Hahn Decomposition

    Absolute Continuity and Singularity

    The Main Theorem

    33. CONDITIONAL PROBABILITY, 454

    The Discrete Case

    The General Case

    Properties of Conditional Probability

    Difficulties and Curiosities

    Conditional Probability Distributions

    34. CONDITIONAL EXPECTATION, 472

    Definition

    Properties of Conditional Expectation

    Conditional Distributions and Expectations

    Sufficient Subfields*

    Minimum-Variance Estimation*

    35. MARTINGALES, 487

    Definition

    Submartingales

    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*

    Chapter7 STOCHASTIC PROCESSES 513

    36. KOLMOGOROV'S EXISTENCE THEOREM, 513

    Stochastic Processes

    Finite-Dimensional Distributions

    Product Spaces

    Kolmogorov's Existence Theorem

    The Inadequacy of RT

    A Return to Ergodic Theory

    The Hewitt-Savage Theorem*

    37. BROWNIAN MOTION, 530

    Definition

    Continuity of Paths

    Measurable Processes

    Irregularity of Brownian Motion Paths

    The Strong Markov Property

    The Reflection Principle

    Skorohod Embedding

    Invariance*

    38. NONDENUMERABLE PROBABILITIES, 558

    Introduction

    Definitions

    Existence Theorems

    Consequences of Separability*

    APPENDIX 571

    NOTES ON THE PROBLEMS 587

    BIBLIOGRAPHY 617

    INDEX 619