概率与测度

FOREWORD
PREFACE
Patrick Billingsley 1925- 2011
Chapter 1
PROBABILITY
1. BOREL'S NORMAL NUMBER THEOREM, 1
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FOREWORD
PREFACE
Patrick Billingsley 1925- 2011
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, 18
Spaces ——Assigning Probabilities——Classes of Sets——Probability
Measures——Lebesgue Measure on the Unit Interval——Sequence
Space* - Constructing σ-Fields*
3. EXISTENCE AND EXTENSION, 39
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, 53
General Formulas—— Limit Sets-Independent
Events——Subfields——The Borel-Cantelelli
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 BoreI-Cantelli
Lemma
7. GAMBLING SYSTEMS, 98
Gambler's Ruin——Selection Systems——Gambling Policies——Bold
Play*——Timid Play*
8. MARKOVCHAINS, 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
Chapter 2
MEASURE 167
10. GENERAL MEASURES, 167
Classes of Sets—— Conventions Involving
∞ —— 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*
Chapter 3
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
Chapter 4
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
Chapter 5
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
Chapter 6
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 —— Su bmartingales—— 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 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 ——I nvariance*
38. NONDENUMERABLE PROBABILITIES, 558
Introduction —— Definitions—— Existence
Theorems——Consequences of Separability*
APPENDIX
NOTES ON THE PROBLEMS
BIBLIOGRAPHY
INDEX
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