A Second Course in Probability

1. Measure Theory and Laws of Large Numbers -- Introduction -- A Non-Measurable Event -- Countable and Uncountable Sets -- Probability Spaces -- Random Variables -- Expected Value -- Almost Sure Convergence and the Dominated Convergence Theorem -- Convergence in Probablitiy and in Distribution -- Law of Large Numbers and Ergodic Theorem -- Exercises -- 2. Stein's Method and Central Limit Theorems -- Introduction -- Coupling -- Poisson Approximation and Le Cam's Theorem -- The Stein-Chen Method -- Stein's Method for the Geometric Distribution -- Stein's Method for the Normal Distribution -- Exercises -- 3. Conditional Expectation and Martingales -- Introduction -- Conditional Expectation -- Martingales -- The Martingale Stopping Theorem -- The Hoeffding-Azuma Inequality -- Submartingales, Supermartingales, and a Convergence Theorem -- Exercises -- 4. Bounding Probabilities and Expectations -- Introduction -- Jensen's Inequality -- Probability Bounds via the Importance Sampling Identity -- Chernoff Bounds -- Second Moment and Conditional Expectation Inequalities -- The Min-Max Identity and Bounds on the Maximum -- Stochastic Orderings -- Exercises -- 5. Markov Chains -- Introduction -- The Transition Matrix -- The Strong Markov Property -- Classification of States -- Stationary and Limiting Distributions -- Time Reversibility -- A Mean Passage Time Bound -- Exercises -- 6. Renewal Theory -- Introduction -- Some Limit Theorems of Renewal Theory -- Renewal Reward Processes -- 6.3.1 Queueing Theory Applications of Renewal Reward Processes -- Blackwell's Theorem -- The Poisson Process -- Exercises -- 7. Brownian Motion -- Introduction -- Continuous Time Martingales -- Construction Brownian Motion -- Embedding Variables in Brownian Motion -- The Central Limit Theorem -- Exercises