1 Transition Functions and Resolvents.- 1. Markov Chains and Transition Functions: Definitions and Basic Properties.- 2. Differentiability Properties of Transition Functions and Significance of the Q-Matrix.- 3. Resolvent Functions and Their Properties.- 4. The Functional-Analytic Setting for Transition Functions and Resolvents.- 5. Feller Transition Functions.- 6. Kendall's Representation of Reversible Transition Functions.- 7. Appendix.- 2 Existence and Uniqueness of Q-Functions.- 1. Q-Functions and the Kolmogorov Backward and Forward Equations.- 2. Existence and Uniqueness of Q-Functions.- 3 Examples of Continuous-Time Markov Chains.- 1. Finite Markov Chains.- 2. Birth and Death Processes.- 3. Continuous Time Parameter Markov Branching Processes.- 4 More on the Uniqueness Problem.- 1. Laplace Transform Tools.- 2. Non-uniqueness-Construction of Q-Functions Other Than the Minimal One.- 3. Uniqueness-The Non-Conservative Case.- 5 Classification of States and Invariant Measures.- 1. Classification of States.- 2. Sub-invariant and Invariant Measures.- 3. Classification Based on the Q-Matrix.- 4. Determination of Invariant Measures from the Q-Matrix.- 6 Strong and Exponential Ergodicity.- 1. The Ergodic Coefficient and Hitting Times.- 2. Ordinary Ergodicity.- 3. Strong Ergodicity.- 4. Geometric Ergodicity for Discrete Time Chains.- 5. The Croft-Kingman Lemmas.- 6. Exponential Ergodicity for Continuous-Time Chains.- 7 Reversibility, Monotonicity, and Other Properties.- 1. Symmetry and Reversibility.- 2. Exponential Families of Transition Functions.- 3. Stochastic Monotonicity and Comparability.- 4. Dual Processes.- 5. Coupling.- 8 Birth and Death Processes.- 1. The Potential Coefficients and Feller's Boundary Conditions.- 2. Karlin and McGregor's Representation Theorem and Duality.- 3. The Stieltjes Moment Problem.- 4. The Karlin-McGregor Method of Solution.- 5. Total Positivity of the Birth and Death Transition Function.- 9 Population Processes.- 1. Upwardly Skip-Free Processes.- 2. Extinction Times and Probability of Extinction for Upwardly Skip-Free Processes.- 3. Multidimensional Population Processes.- 4. Two-Dimensional Competition Processes.- 5. Birth, Death, and Migration Processes.- Symbol Index.- Author Index.
Continuous time parameter Markov chains have been useful for modeling various random phenomena occurring in queueing theory, genetics, demography, epidemiology, and competing populations. This is the first book about those aspects of the theory of continuous time Markov chains which are useful in applications to such areas. It studies continuous time Markov chains through the transition function and corresponding q-matrix, rather than sample paths. An extensive discussion of birth and death processes, including the Stieltjes moment problem, and the Karlin-McGregor method of solution of the birth and death processes and multidimensional population processes is included, and there is an extensive bibliography. Virtually all of this material is appearing in book form for the first time.
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