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To some extent, it would be accurate to summarize the contents of this book as an intolerably protracted description of what happens when either one raises a transition probability matrix P (i. e. , all entries (P)»j are n- negative and each row of P sums to 1) to higher and higher powers or one exponentiates R(P — I), where R is a diagonal matrix with non-negative entries. Indeed, when it comes right down to it, that is all that is done in this book. However, I, and others of my ilk, would take offense at such a dismissive characterization of the theory of Markov chains and processes with values in a countable state space, and a primary goal of mine in writing this book was to convince its readers that our offense would be warranted. The reason why I, and others of my persuasion, refuse to consider the theory here as no more than a subset of matrix theory is that to do so is to ignore the pervasive role that probability plays throughout. Namely, probability theory provides a model which both motivates and provides a context for what we are doing with these matrices. To wit, even the term "transition probability matrix" lends meaning to an otherwise rather peculiar set of hypotheses to make about a matrix.

Markov processes. --- Stochastic processes. --- Random processes --- Probabilities --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Stochastic processes --- Distribution (Probability theory. --- Probability Theory and Stochastic Processes. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk

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Stochastic processes --- Probabilities --- Probabilités --- 519.2 --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Probability. Mathematical statistics --- Probabilities. --- 519.2 Probability. Mathematical statistics --- Probabilités

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Integralen [Veralgemeende ] --- Integrales generalisees --- Integrals [Generalized ] --- Measure theory --- Mesure [Théorie de la ] --- Metingen--Theorie --- Integrals, Generalized --- Intégrales généralisées --- Mesure, Théorie de la --- 517.2 --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Measure algebras --- Rings (Algebra) --- Calculus, Integral --- Differential calculus. Differentiation --- 517.2 Differential calculus. Differentiation --- Intégrales généralisées --- Mesure, Théorie de la --- Integrals, Generalized. --- Measure theory.

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This book provides a rigorous but elementary introduction to the theory of Markov Processes on a countable state space. It should be accessible to students with a solid undergraduate background in mathematics, including students from engineering, economics, physics, and biology. Topics covered are: Doeblin's theory, general ergodic properties, and continuous time processes. Applications are dispersed throughout the book. In addition, a whole chapter is devoted to reversible processes and the use of their associated Dirichlet forms to estimate the rate of convergence to equilibrium. These results are then applied to the analysis of the Metropolis (a.k.a simulated annealing) algorithm. The corrected and enlarged 2nd edition contains a new chapter in which the author develops computational methods for Markov chains on a finite state space. Most intriguing is the section with a new technique for computing stationary measures, which is applied to derivations of Wilson's algorithm and Kirchoff's formula for spanning trees in a connected graph.

Distribution (Probability theory) --- Markov processes. --- Markov, Processus de. --- Markov-processen. --- Processos de markov. --- Markov-Prozess. --- Mathematics. --- Differentiable dynamical systems. --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Distribution functions --- Frequency distribution --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Math --- Dynamics. --- Ergodic theory. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Dynamical Systems and Ergodic Theory. --- Characteristic functions --- Probabilities --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Distribution (Probability theory. --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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Stochastic processes --- 519.217 --- 519.217 Markov processes --- Markov processes --- Diffusion processes