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Chapter 1. Some Background and Preliminaries
1.1 The Language of Probability Theory
1.1.1. Sample Spaces and Events
1.1.2. Probability Measures
Exercises for 1.1
1.2 Finite and Countable Sample Spaces
1.2.1. Probability Theory on a Countable Space
1.2.2. Uniform Probabilities and Coin Tossing
1.2.3. Tournaments
1.2.4. Symmetric Random Walk
1.2.5. De Moivre’s Central Limit Theorem
1.2.6. Independent Events
1.2.7. The Arc Sine Law
1.2.8. Conditional Probability
Exercises for 1.2
1.3 Some Non-Uniform Probability Measures
1.3.1. Random Variables and Their Distributions
1.3.2. Biased Coins
1.3.3. Recurrence and Transience of Random Walks
Exercises for 1.3
1.4 Expectation Values
1.4.1. Some Elementary Examples
1.4.2. Independence and Moment Generating Functions
1.4.3. Basic Convergence Results
Exercises for 1.4
Comments on Chapter 1
Chapter 2. Probability Theory on Uncountable Sample Spaces
2.1 A Little Measure Theory
2.1.1. Sigma Algebras, Measurable Functions, and Measures
2.1.2. Π- and Λ-Systems
Exercises for 2.1
2.2 A Construction of Pp on {0, 1}Z+
2.2.1. The Metric Space {0, 1}Z+
2.2.2. The Construction
Exercises for 2.2
2.3 Other Probability Measures
2.3.1. The Uniform Probability Measure on [0, 1]
2.3.2. Lebesgue Measure on R
2.3.3. Distribution Functions and Probability Measures
Exercises for 2.3
2.4 Lebesgue Integration
2.4.1. Integration of Functions
2.4.2. Some Properties of the Lebesgue Integral
2.4.3. Basic Convergence Theorems
2.4.4. Inequalities
2.4.5. Fubini’s Theorem
Exercises for 2.4
2.5 Lebesgue Measure on RN
2.5.1. Polar Coordinates
2.5.2. Gaussian Computations and Stirling’s Formula
Exercises for 2.5
Comments on Chapter 2
Chapter 3. Some Applications to Probability Theory
3.1 Independence and Conditioning
3.1.1. Independent σ-Algebras
3.1.2. Independent Random Variables
3.1.3. Conditioning
3.1.4. Some Properties of Conditional Expectations
Exercises for 3.1
3.2 Distributions that Admit a Density
3.2.1. Densities
3.2.2. Densities and Conditioning
Exercises for 3.2
3.3 Summing Independent Random Variables
3.3.1. Convolution of Distributions
3.3.2. Some Important Examples
3.3.3. Kolmogorov’s Inequality and the Strong Law
Exercises for 3.3
Comments on Chapter 3
Chapter 4. The Central Limit Theorem and Gaussian Distributions
4.1 The Central Limit Theorem
4.1.1. Lindeberg’s Theorem
Exercises for 4.1
4.2 Families of Normal Random Variables
4.2.1. Multidimensional Gaussian Distributions
4.2.2. Standard Normal Random Variables
4.2.3. More General Normal Random Variables
4.2.4. A Concentration Property of Gaussian Distributions
4.2.5. Linear Transformations of Normal Random Variables
4.2.6. Gaussian Families
Exercises for 4.2
Comments on Chapter 4
Chapter 5. Discrete Parameter Stochastic Processes
5.1 Random Walks Revisited
5.1.1. Immediate Rewards
5.1.2. Computations via Conditioning
Exercises for 5.1
5.2 Processes with the Markov Property
5.2.1. Sequences of Dependent Random Variables
5.2.2. Markov Chains
5.2.3. Long-Time Behavior
5.2.4. An Extension
Exercises for 5.2
5.3 Markov Chains on a Countable State Space
5.3.1. The Markov Property
5.3.2. Return Times and the Renewal Equation
5.3.3. A Little Ergodic Theory
Exercises for 5.3
Comments on Chapter 5
Chapter 6. Some Continuous-Time Processes
6.1 Transition Probability Functions and Markov Processes
6.1.1. Transition Probability Functions
Exercises for 6.1
6.2 Markov Chains Run with a Poisson Clock
6.2.1. The Simple Poisson Process
6.2.2. A Generalization
6.2.3. Stationary Measures
Exercises for 6.2
6.3 Brownian Motion
6.3.1. Some Preliminaries
6.3.2. L´evy’s Construction
6.3.3. Some Elementary Properties of Brownian Motion
6.3.4. Path Properties
6.3.5. The Ornstein–Uhlenbeck Process
Exercises for 6.3
Comments on Chapter 6
Chapter 7. Martingales
7.1 Discrete Parameter Martingales
7.1.1. Doob’s Inequality
Exercises for 7.1
7.2 The Martingale Convergence Theorem
7.2.1. The Convergence Theorem
7.2.2. Application to the Radon–Nikodym Theorem
Exercises for 7.2
7.3 Stopping Times
7.3.1. Stopping Time Theorems
7.3.2. Reversed Martingales
7.3.3. Exchangeable Sequences
Exercises for 7.3
7.4 Continuous Parameter Martingales
7.4.1. Progressively Measurable Functions
7.4.2. Martingales and Submartingales
7.4.3. Stopping Times Again
7.4.4. Continuous Martingales and Brownian Motion
7.4.5. Brownian Motion and Differential Equations
Exercises for 7.4
Comments on Chapter 7
Notation
Bibliography
Index
