Preface
Table of Dependence
Chapter 1 Sums of Independent Random Variables
1.1 Independence
1.1.1.Independent σ—Algebras
1.1.2.Independent Functions
1.1.3.The Rademacher Functions
Exercises for 1.1
1.2 The Weak Law of Large Numbers
1.2.1.Orthogonal Random Variables
1.2.2.Independent Random Variables
1.2.3.Approximate Identities
Exercises for 1.2
1.3 Cramer's Theory of Large Deviations
Exercises for 1.3
1.4 The Strong Law of Large Numbers
Exercises for 1.4
1.5 Law of the Iterated Logarithm
Exercises for 1.5
Chapter 2 The Central Limit Theorem
2.1 The Basic Central Limit Theorem
2.1.1.Lindeberg's Theorem
2.1.2.The Central Limit Theorem
Exercisesfor 2.1
2.2 The Berry—Esseen Theorem via Stein's Method
2.2.1.L1—Berry—Esseen
2.2.2.The Classical Berry—Esseen Theorem
Exercises for 2.2
2.3 Some Extensions of The Central Limit Theorem
2.3.1.The Fourier Transform
2.3.2.Multidimensional Central Limit Theorem
2.3.3.Higher Moments
Exercises for 2.3
2.4 An Application to Hermite Multipliers
2.4.1.Hermite Multipliers
2.4.2.Beckner's Theorem
2.4.3.Applications of Beckner's Theorem
Exercises for 2.4
Chapter 3 Infinitely Divisible Laws
3.1 Convergence of Measures on RN
3.1.1.Sequential Compactnessin M1（RN）
3.1.2.Levy's Continuity Theorem
Exercises for 3.1
3.2 The Levy—Khinchine Formula
3.2.1.T（RN） Is the Closure of P（RN）
3.2.2.The Formula
Exercises for 3.2
3.3 Stable Laws
3.3.1.General Results
3.3.2.α—Stable Laws
Exercises for 3.3
Chapter 4 Levy Processes
4.1 Stochastic Processes， Some Generalities
4.1.1.The Space D（RN）
4.1.2.Jump Functions
Exercises for 4.1
4.2 Discontinuous Levy Processes
4.2.1.The Simple Poisson Process
4.2.2.Compound Poisson Processes
4.2.3.Poisson Jump Processes
4.2.4.Levy Processes with Bounded Variation
4.2.5.General， Non—Gaussian Levy Processes
Exercises for 4.2
4.3 Brownian Motion， the Gaussian Levy Process
4.3.1.Deconstructing Brownian Motion
4.3.2.Levy's Construction of Brownian Motion
4.3.3.Levy's Constructionin Context
4.3.4.Brownian Paths Are Non—Differentiable
4.3.5.General Levy Processes
Exercises for 4.3
Chapter 5 Conditioning and Martingales
5.1 Conditioning
5.1.1.Kolmogorov's Definition
5.1.2.Some Extensions
Exercises for 5.1
5.2 Discrete Parameter Martingales
5.2.1.Doob's Inequality and Marcinkewitz's Theorem
5.2.2.Doob's Stopping Time Theorem
5.2.3.Martingale Convergence Theorem
5.2.4.Reversed Martingales and De Finetti's Theory
5.2.5.An Application to a Tracking Algorithm
Exercises for 5.2
Chapter 6 Some Extensions and Applications of Martingale Theory
6.1 Some Extensions
6.1.1.Martingale Theory for a σ—Finite Measure Space
6.1.2.Banach Space—Valued Martingales
Exercises for 6.1
6.2 Elements of Ergodic Theory
6.2.1.The Maximal Ergodic Lemma
6.2.2.Birkhoff's Ergodic Theorem
6.2.3.Stationary Sequences
6.2.4.Continuous Parameter Ergodic Theory
Exercises for 6.2
6.3 Burkholder's Inequality
6.3.1.Burkholder's Comparison Theorem
6.3.2.Burkholder'slnequality
Exercises for 6.3
Chapter 7 Continuous Parameter Martingales
7.1 Continuous Parameter Martingales
7.1.1.Progressively Measurable Functions
7.1.2.Martingales： Definition and Examples
7.1.3.Basic Results
7.1.4.Stopping Times and Stopping Theorems
7.1.5.An Integration by Parts Formula
Exercises for 7.1
7.2 Brownian Motion and Martingales
7.2.1.Levy's Characterization of Brownian Motion
7.2.2.Doob—Meyer Decomposition， an Easy Case
7.2.3.Burkholder's Inequality Again
Exercises for 7.2
7.3 The Refiection Principle Revisited
7.3.1.Reflecting Symmetric Levy Processes
7.3.2.Reflected Brownian Motion
Exercises for 7.3
Chapter 8 Gaussian Measures on a Banach Space
8.1 The Classical Wiener Space
8.1.1.Classical Wiener Measure
8.1.2.The Classical Cameron—Martin Space
Exercisesfor 8.1
8.2 A Structure Theorem for Gaussian Measures
8.2.1.Fernique's Theorem
8.2.2.The Basic Structure Theorem
8.2.3.The Cameron—Marin Space
Exercises for 8.2
8.3 From Hilbert to Abstract Wiener Space
8.3.1.An Isomorphism Theorem
8.3.2.Wiener Series
8.3.3.Orthogonal Projections
8.3.4.Pinned Brownian Motion
8.3.5.Orthogonal Invariance
Exercises for 8.3
8.4 A Larger Deviations Result and Strassen's Theorem
8.4.1.Large Deviations for Abstract Wiener Space
8.4.2.Strassen's Law ofthelterated Logarithm
Exercises for 8.4
8.5 Euclidean Free Fields
8.5.1.The Ornstein—Uhlenbeck Process
8.5.2.Ornstein—Uhlenbeck as an Abstract Wiener Space
8.5.3.Higher Dimensional Free Fields
Exercises for 8.5
8.6 Brownian Motion on a Banach Space
8.6.1.Abstract Wiener Formulation
8.6.2.Brownian Formulation
8.6.3.Strassen's Theorem Revisited
Exercises for 8.6
Chapter 9 Convergence of Measures on a Polish Space
9.1 Prohorov—Varadarajan Theory
9.1.1.Some Background
9.1.2.The Weak Topology
9.1.3.The Levy Metric and Completeness of M1（E）
Exercises for 9.1
9.2 Regular Conditional Probability Distributions
9.2.1.Fibering a Measure
9.2.2.Representing Levy Measures via the Ito Map
Exercises for 9.2
9.3 Donsker'slnvariance Principle
9.3.1.Donsker's Theorem
9.3.2.Rayleigh's Random Flights Model
Exercise for 9.3
Chapter 10 Wiener Measure and Partial Differential Equations
10.1 Martingales and Partial Differential Equations
10.1.1. Localizing and Extending Martingale Representations
10.1.2. Minimum Principles
10.1.3. The Hermite Heat Equation
10.1.4. The Arcsine Law
10.1.5. Recurrence and Transience of Brownian Motion
Exercises for 10.1
10.2 The Markov Property and Potential Theory
10.2.1. The Markov Property for Wiener Measure
10.2.2. Recurrence in One and Two Dimensions
10.2.3. The Dirichlet Problem
Exercises for 10.2
10.3 Other Heat Kernels
10.3.1. A General Construction
10.3.2. The Dirichlet Heat Kernel
10.3.3. Feynman-Kac Heat Kernels
10.3.4. Ground States and Associated Measures on Pathspace
10.3.5. Producing Ground States
Exercises for 10.3
Chapter 11 Some Classical Potential Theory
11.1 Uniqueness Refined
11.1.1. The Dirichlet Heat Kernel Again
11.1.2. Exiting Through Oreg G
11.1.3. Applications to Questions of Uniqueness
11.1.4. Harmonic Measure
Exercises for 11.1
11.2 The Poisson Problem and Green Functions
11.2.1. Green Functions when N > 3
11.2.2. Green Functions when N E {1, 2}
Exercises for 11.2
11.3 Excessive Functions, Potentials, and Riesz Decompositions
11.3.1. Excessive Functions
11.3.2. Potentials and Riesz Decomposition
Exercises for 11.3
11.4 Capacity
11.4.1. The Capacitory Potential
11.4.2. The Capacitory Distribution
11.4.3. Wiener's Test
11.4.4. Some Asymptotic Expressions Involving Capacity
Exercises for 11.4
Notation
Index
