Prelude: Random Walks. - The Model. - Issues and Approaches. - Functional of the Random Walk. - Limit Theorems. - Summary. - 1 Probability. - 1. 1 Random Experiments and Sample Spaces. - 1. 2 Events and Classes of Sets. - 1. 3 Probabilities and Probability Spaces. - 1. 4 Probabilities on R. - 1. 5 Conditional Probability Given a Set. - 1. 6 Complements. - 1. 7 Exercises. - 2 Random Variables. - 2. 1 Fundamentals. - 2. 2 Combining Random Variables. - 2. 3 Distributions and Distribution Functions. - 2. 4 Key Random Variables and Distributions. - 2. 5 Transformation Theory. - 2. 6 Random Variables with Prescribed Distributions. - 2. 7 Complements. - 2. 8 Exercises. - 3 Independence. - 3. 1 Independent Random Variables. - 3. 2 Functions of Independent Random Variables. - 3. 3 Constructing Independent Random Variables. - 3. 4 Independent Events. - 3. 5 Occupancy Models. - 3. 6 Bernoulli and Poisson Processes. - 3. 7 Complements. - 3. 8 Exercises. - 4 Expectation. - 4. 1 Definition and Fundamental Properties. - 4. 2 Integrals with respect to Distribution Functions. - 4. 3 Computation of Expectations. - 4. 4 LP Spaces and Inequalities. - 4. 5 Moments. - 4. 6 Complements. - 4. 7 Exercises. - 5 Convergence of Sequences of Random Variables. - 5. 1 Modes of Convergence. - 5. 2 Relationships Among the Modes. - 5. 3 Convergence under Transformations. - 5. 4 Convergence of Random Vectors. - 5. 5 Limit Theorems for Bernoulli Summands. - 5. 6 Complements. - 5. 7 Exercises. - 6 Characteristic Functions. - 6. 1 Definition and Basic Properties. - 6. 2 Inversion and Uniqueness Theorems. - 6. 3 Moments and Taylor Expansions. - 6. 4 Continuity Theorems and Applications. - 6. 5 Other Transforms. - 6. 6 Complements. - 6. 7 Exercises. - 7 Classical Limit Theorems. - 7. 1 Series of Independent Random Variables. - 7. 2 The Strong Law of Large Numbers. - 7. 3 The Central Limit Theorem. - 7. 4 The Law ofthe Iterated Logarithm. - 7. 5 Applications of the Limit Theorems. - 7. 6 Complements. - 7. 7 Exercises. - 8 Prediction and Conditional Expectation. - 8. 1 Prediction in L2. - 8. 2 Conditional Expectation Given a Finite Set of Random Variables. - 8. 3 Conditional Expectation for X?L2. - 8. 4 Positive and Integrable Random Variables. - 8. 5 Conditional Distributions. - 8. 6 Computational Techniques. - 8. 7 Complements. - 8. 8 Exercises. - 9 Martingales. - 9. 1 Fundamentals. - 9. 2 Stopping Times. - 9. 3 Optional Sampling Theorems. - 9. 4 Martingale Convergence Theorems. - 9. 5 Applications of Convergence Theorems. - 9. 6 Complements. - 9. 7 Exercises. - A Notation. - B Named Objects. Language: English