Title | Markov Chain Models โ Rarity and Exponentiality [electronic resource] / edited by Julian Keilson |
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Imprint | New York, NY : Springer New York, 1979 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-6200-8 |
Descript | XIV, 184 p. online resource |
0. Introduction and Summary -- 1. Discrete Time Markov Chains; Reversibility in Time -- ยง1.00. Introduction -- ยง1.0. Notation, Transition Laws -- ยง1.1. Irreducibility, Aperiodicity, Ergodicity; Stationary Chains -- ยง1.2. Approach to Ergodicity; Spectral Structure, Perron-Romanovsky-Frobenius Theorem -- ยง1.3. Time-Reversible Chains -- 2. Markov Chains in Continuous Time; Uniformization; Reversibility -- ยง2.00. Introduction -- ยง2.0. Notation, Transition Laws; A Review -- ยง2.1. Uniformizable Chains โ A Bridge Between Discrete and Continuous Time Chains -- ยง2.2. Advantages and Prevalence of Uniformizable Chains -- ยง2.3. Ergodicity for Continuous Time Chains -- ยง2.4. Reversibility for Ergodic Markov Chains in Continuous Time -- ยง2.5. Prevalence of Time-Reversibility -- 3. More on Time-Reversibility; Potential Coefficients; Process Modification -- ยง3.00. Introduction -- ยง3.1. The Advantages of Time-Reversibility -- ยง3.2. The Spectral Representation -- ยง3.3. Potentials; Spectral Representation -- ยง3.4. More General Time-Reversible Chains -- ยง3.5. Process Modifications Preserving Reversibility -- ยง3.6. Replacement Processes -- 4. Potential Theory, Replacement, and Compensation -- ยง4.00. Introduction -- ยง4.1. The Green Potential -- ยง4.2. The Ergodic Distribution for a Replacement Process -- ยง4.3. The Compensation Method -- ยง4.4. Notation for the Homogeneous Random Walk -- ยง4.5. The Compensation Method Applied to the Homogeneous Random Walk Modified by Boundaries -- ยง4.6. Advantages of the Compensation Method. An Illustrative Example -- ยง4.7. Exploitation of the Structure of the Green Potential for the Homogeneous Random Walk -- ยง4.8. Similar Situations -- 5. Passage Time Densities in Birth-Death Processes; Distribution Structure -- ยง5.00. Introduction -- ยง5.1. Passage Time Densities for Birth-Death Processes -- ยง5.2. Passage Time Moments for a Birth-Death Process -- ยง5.3. PF?, Complete Monotonicity, Log-Concavity and Log-Convexity -- ยง5.4. Complete Monotonicity and Log-Convexity -- ยง5.5. Complete Monotonicity in Time-Reversible Processes -- ยง5.6. Some Useful Inequalities for the Families CM and PF? -- ยง5.7. Log-Concavity and Strong Unimodality for Lattice Distributions -- ยง5.8. Preservation of Log-Concavity and Log-Convexity under Tail Summation and Integration -- ยง5.9. Relation of CM and PF? to IFR and DFR Classes in Reliability -- 6. Passage Times and Exit Times for More General Chains -- ยง6.00. Introduction -- ยง6.1. Passage Time Densities to a Set of States -- ยง6.2. Mean Passage Times to a Set via the Green Potential -- ยง6.3. Ruin Probabilities via the Green Potential -- ยง6.4. Ergodic Flow Rates in a Chain -- ยง6.5. Ergodic Exit Times, Ergodic Sojourn Times, and Quasi-Stationary Exit Times -- ยง6.6. The Quasi-Stationary Exit Time. A Limit Theorem -- ยง6.7. The Connection Between Exit Times and Sojourn Times. A Renewal Theorem -- ยง6.8. A Comparison of the Mean Ergodic Exit Time and Mean Ergodic Sojourn Time for Arbitrary Chains -- ยง6.9. Stochastic Ordering of Exit Times of Interest for Time-Reversible Chains -- ยง6.10. Superiority of the Exit Time as System Failure Time; Jitter -- 7. The Fundamental Matrix, and Allied Topics -- ยง7.00. Introduction -- ยง7.1. The Fundamental Matrix for Ergodic Chains -- ยง7.2. The Structure of the Fundamental Matrix for Time-Reversible Chains -- ยง7.3. Mean Failure Times and Ruin Probabilities for Systems with Independent Markov Components and More General Chains -- ยง7.4. Covariance and Spectral Density Structure for Time-Reversible Processes -- ยง7.5. A Central Limit Theorem -- ยง7.6. Regeneration Times and Passage Times-Their Relation For Arbitrary Chains -- ยง7.7. Passage to a Set with Two States -- 8. Rarity and Exponentiality -- ยง8.0. Introduction -- ยง8.1. Passage Time Density Structure for Finite Ergodic Chains; the Exponential Approximation -- ยง8.2. A Limit Theorem for Ergodic Regenerative Processes -- ยง8.3. Prototype Behavior: Birth-Death Processes; Strongly Stable Systems -- ยง8.4. Limiting Behavior of the Ergodic and Quasi-stationary Exit Time Densities and Sojourn Time Densities for Birth-Death Processes -- ยง8.5. Limit Behavior of Other Exit Times for More General Chains -- ยง8.6. Strongly Stable Chains, Jitter; Estimation of the Failure Time Needed for the Exponential Approximation -- ยง8.7. A Measure of Exponentiality in the Completely Monotone Class of Densities -- ยง8.8. An Error Bound for Departure from Exponentiality in the Completely Monotone Class -- ยง8.9. The Exponential Approximation for Time-Reversible Systems -- ยง8.10. A Relaxation Time of Interest -- 9. Stochastic Monotonicity -- ยง9.00. Introduction -- ยง9.1. Monotone Markov Matrices and Monotone Chains -- ยง9.2. Some Monotone Chains in Discrete Time -- ยง9.3. Monotone Chains in Continuous Time -- ยง9.4. Other Monotone Processes in Continuous Time -- References