Author | Kemeny, John G. author |
---|---|
Title | Denumerable Markov Chains [electronic resource] : with a chapter of Markov Random Fields by David Griffeath / by John G. Kemeny, J. Laurie Snell, Anthony W. Knapp |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1976 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-9455-6 |
Descript | XII, 484 p. online resource |
1: Prerequisites from Analysis -- 1. Denumerable Matrices -- 2. Measure Theory -- 3. Measurable Functions and Lebesgue Integration -- 4. Integration Theorems -- 5. Limit Theorems for Matrices -- 6. Some General Theorems from Analysis -- 2: Stochastic Processes -- 1. Sequence Spaces -- 2. Denumerable Stochastic Processes -- 3. Borel Fields in Stochastic Processes -- 4. Statements of Probability Zero or One -- 5. Conditional Probabilities -- 6. Random Variables and Means -- 7. Means Conditional on Statements -- 8. Problems -- 3: Martingales -- 1. Means Conditional on Partitions and Functions -- 2. Properties of Martingales -- 3. A First Martingale Systems Theorem -- 4. Martingale Convergence and a Second Systems Theorem -- 5. Examples of Convergent Martingales -- 6. Law of Large Numbers -- 7. Problems -- 4: Properties of Markov Chains -- 1. Markov Chains -- 2. Examples of Markov Chains -- 3. Applications of Martingale Ideas -- 4. Strong Markov Property -- 5. Systems Theorems for Markov Chains -- 6. Applications of Systems Theorems -- 7. Classification of States -- 8. Problems -- 5: Transient Chains -- 1. Properties of Transient Chains -- 2. Superregular Functions -- 3. Absorbing Chains -- 4. Finite Drunkardโs Walk -- 5. Infinite Drunkardโs Walk -- 6. A Zero-One Law for Sums of Independent Random Variables -- 7. Sums of Independent Random Variables on the Line -- 8. Examples of Sums of Independent Random Variables -- 9. Ladder Process for Sums of Independent Random Variables -- 10. The Basic Example -- 11. Problems -- 6: Recurrent Chains -- 1. Mean Ergodic Theorem for Markov Chains -- 2. Duality -- 3. Cyclicity -- 4. Sums of Independent Random Variables -- 5. Convergence Theorem for Noncyclic Chains -- 6. Mean First Passage Time Matrix -- 7. Examples of the Mean First Passage Time Matrix -- 8. Reverse Markov Chains -- 9. Problems -- 7: Introduction to Potential Theory -- 1. Brownian Motion -- 2. Potential Theory -- 3. Equivalence of Brownian Motion and Potential Theory -- 4. Brownian Motion and Potential Theory in n Dimensions -- 5. Potential Theory for Denumerable Markov Chains -- 6. Brownian Motion as a Limit of the Symmetric Random Walk -- 7. Symmetric Random Walk in n Dimensions -- 8: Transient Potential Theory -- 1. Potentials -- 2. The h-Process and Some Applications -- 3. Equilibrium Sets and Capacities -- 4. Potential Principles -- 5. Energy -- 6. The Basic Example -- 7. An Unbounded Potential -- 8. Applications of Potential-Theoretic Methods -- 9. General Denumerable Stochastic Processes -- 10. Problems -- 9: Recurrent Potential Theory -- 1. Potentials -- 2. Normal Chains -- 3. Ergodic Chains -- 4. Classes of Ergodic Chains -- 5. Strong Ergodic Chains -- 6. The Basic Example -- 7. Further Examples -- 8. The Operator K -- 9. Potential Principles -- 10. A Model for Potential Theory -- 11. A Nonnormal Chain and Other Examples -- 12. Two-Dimensional Symmetric Random Walk -- 13. Problems -- 10: Transient Boundary Theory -- 1. Motivation for Martin Boundary Theory -- 2. Extended Chains -- 3. Martin Exit Boundary -- 4. Convergence to the Boundary -- 5. Poisson-Martin Representation Theorem -- 6. Extreme Points of the Boundary -- 7. Uniqueness of the Representation -- 8. Analog of Fatouโs Theorem -- 9. Fine Boundary Functions -- 10. Martin Entrance Boundary -- 11. Application to Extended Chains -- 12. Proof of Theorem 10.9 -- 13. Examples -- 14. Problems -- 11: Recurrent Boundary Theory -- 1. Entrance Boundary for Recurrent Chains -- 2. Measures on the Entrance Boundary -- 3. Harmonic Measure for Normal Chains -- 4. Continuous and T-Continuous Functions -- 5. Normal Chains and Convergence to the Boundary -- 6. Representation Theorem -- 7. Sums of Independent Random Variables -- 8. Examples -- 9. Problems -- 12: Introduction to Random Fields -- 1 Markov Fields -- 2. Finite Gibbs Fields -- 3. Equivalence of Finite Markov and Neighbor Gibbs Fields -- 4. Markov Fields and Neighbor Gibbs Fields: the Infinite Case -- 5. Homogeneous Markov Fields on the Integers -- 6. Examples of Phase Multiplicity in Higher Dimensions -- 7. Problems -- Notes -- Additional Notes -- References -- Additional References -- Index of Notation