FOREWORD ....................................................... xi
PREFACE ...................................................... xiii
Patrick Billingsley 1925-2011 .................................. xv
Chapter 1 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 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-Cantelli Lemmas - The Zero - One Law
5 SIMPLE RANDOM VARIABLES ..................................... 72
Definition - Convergence of Random Variables -
Independence - Existence of Independent Sequences -
Expected Value - Inequalities Asterisks indicate topics
that may be omitted on a first reading.
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
Chapter 2 MEASURE ........................................... 167
10 GENERAL MEASURES ........................................... 167
Classes of Sets-Conventions Involving oo-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 - Subf ields -
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 - 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
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  T - 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
| 
