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"Population processes are stochastic models for systems involving a number of similar particles. Examples include models for chemical reactions and for epidemics. The model may involve a finite number of attributes, or even a continuum. This monograph considers approximations that are possible when the number of particles is large. The models considered will involve a finite number of different types of particles."

"Hidden Markov models (HMMs) are discrete-state, discrete-time, stochastic dynamical systems. They are often used to approximate systems with continuous state spaces operating in continuous time. In addition to introducing the basic ideas of HMMs and algorithms for using them, this book explains the derivations of the algorithms with enough supporting theory to enable readers to develop their own variants. The book also presents Kalman filtering as an extension of ideas from basic HMMs to models with continuous state spaces.Although applications of HMMs have become numerous (396,000 Google hits) since they emerged as the key technology for speech recognition in the 1980s, no introductory book on HMMs in general is available. This text aims to fill that gap.Hidden Markov Models and Dynamical Systems features illustrations that use the Lorenz system, laser data, and natural language data. The concluding chapter presents the application of HMMs to detecting sleep apnea in experimentally measured electrocardiograms. Algorithms are given in pseudocode in the text, and a working implementation of each algorithm is available on the accompanying website."

"Kronecker products are used to define the underlying Markov chain (MC) in various modeling formalisms, including compositional Markovian models, hierarchical Markovian models, and stochastic process algebras. The motivation behind using a Kronecker structured representation rather than a flat one is to alleviate the storage requirements associated with the MC. With this approach, systems that are an order of magnitude larger can be analyzed on the same platform. The developments in the solution of such MCs are reviewed from an algebraic point of view and possible areas for further research are indicated with an emphasis on preprocessing using reordering, grouping, and lumping and numerical analysis using block iterative, preconditioned projection, multilevel, decompositional, and matrix analytic methods. Case studies from closed queueing networks and stochastic chemical kinetics are provided to motivate decompositional and matrix analytic methods, respectively."

"The present volume contains the most advanced theories on the martingale approach to central limit theorems. Using the time symmetry properties of the Markov processes, the book develops the techniques that allow us to deal with infinite dimensional models that appear in statistical mechanics and engineering (interacting particle systems, homogenization in random environments, and diffusion in turbulent flows, to mention just a few applications). The first part contains a detailed exposition of the method, and can be used as a text for graduate courses. The second concerns application to exclusion processes, in which the duality methods are fully exploited. The third part is about the homogenization of diffusions in random fields, including passive tracers in turbulent flows (including the superdiffusive behavior)."