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The book provides a general introduction to the theory of large deviations and a wide overview of the metastable behaviour of stochastic dynamics. With only minimal prerequisites, the book covers all the main results and brings the reader to the most recent developments. Particular emphasis is given to the fundamental Freidlin-Wentzell results on small random perturbations of dynamical systems. Metastability is first described on physical grounds, following which more rigorous approaches to its description are developed. Many relevant examples are considered from the point of view of the so-called pathwise approach. The first part of the book develops the relevant tools including the theory of large deviations which are then used to provide a physically relevant dynamical description of metastability. Written to be accessible to graduate students, this book provides an excellent route into contemporary research.
Large deviations. --- Stability. --- Dynamics --- Mechanics --- Motion --- Vibration --- Benjamin-Feir instability --- Equilibrium --- Deviations, Large --- Limit theorems (Probability theory) --- Statistics --- Mathematical statistics --- Large deviations --- Stability
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Large deviations
Probability theory --- Large deviations. --- Limit theorems (Probability theory) --- 519.21 --- 519.22 --- Large deviations --- Deviations, Large --- Statistics --- 519.22 Statistical theory. Statistical models. Mathematical statistics in general --- Statistical theory. Statistical models. Mathematical statistics in general --- 519.21 Probability theory. Stochastic processes --- Probability theory. Stochastic processes --- Probabilities
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From the reviews: "... Besides the fact that the author's treatment of large deviations is a nice contribution to the literature on the subject, his book has the virue that it provides a beautifully unified and mathematically appealing account of certain aspects of statistical mechanics. ... Furthermore, he does not make the mistake of assuming that his mathematical audience will be familiar with the physics and has done an admireable job of explaining the necessary physical background. Finally, it is clear that the author's book is the product of many painstaking hours of work; and the reviewer is confident that its readers will benefit from his efforts." D. Stroock in Mathematical Reviews 1985 "... Each chapter of the book is followed by a notes section and by a problems section. There are over 100 problems, many of which have hints. The book may be recommended as a text, it provides a completly self-contained reading ..." S. Pogosian in
Large deviations.
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Statistical mechanics.
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Entropy.
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Grandes déviations
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Mécanique statistique
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Entropie
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Electronic books. -- local.
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Large deviations
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Statistical mechanics
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Entropy
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Atomic Physics
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Mathematical Statistics
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Physics
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Mathematics
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Physical Sciences & Mathematics
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536.75
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531.19
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Entropy. Statistical thermodynamics. Irreversible processes
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531.19 Statistical mechanics
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536.75 Entropy. Statistical thermodynamics. Irreversible processes
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Deviations, Large
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Mathematics.
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System theory.
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Probabilities.
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Probability Theory and Stochastic Processes.
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Complex Systems.
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Statistical Physics and Dynamical Systems.
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Mechanics
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Mechanics, Analytic
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Quantum statistics
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Statistical physics
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Thermodynamics
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Limit theorems (Probability theory)
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Statistics
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Distribution (Probability theory.
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Statistical physics.
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Mathematical statistics
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Distribution functions
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Frequency distribution
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Characteristic functions
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Probabilities
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Statistical methods
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Dynamical systems.
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Dynamical systems
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Kinetics
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Force and energy
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Statics
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Probability
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Statistical inference
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Combinations
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Chance
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Least squares
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Risk
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Gaussian processes can be viewed as a far-reaching infinite-dimensional extension of classical normal random variables. Their theory presents a powerful range of tools for probabilistic modelling in various academic and technical domains such as Statistics, Forecasting, Finance, Information Transmission, Machine Learning - to mention just a few. The objective of these Briefs is to present a quick and condensed treatment of the core theory that a reader must understand in order to make his own independent contributions. The primary intended readership are PhD/Masters students and researchers working in pure or applied mathematics. The first chapters introduce essentials of the classical theory of Gaussian processes and measures with the core notions of reproducing kernel, integral representation, isoperimetric property, large deviation principle. The brevity being a priority for teaching and learning purposes, certain technical details and proofs are omitted. The later chapters touch important recent issues not sufficiently reflected in the literature, such as small deviations, expansions, and quantization of processes. In university teaching, one can build a one-semester advanced course upon these Briefs.
Gaussian processes. --- Gaussian processes --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Gaussian measures. --- Isoperimetric inequalities. --- Large deviations. --- Deviations, Large --- Measures, Gaussian --- Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Limit theorems (Probability theory) --- Statistics --- Geometry, Plane --- Inequalities (Mathematics) --- Measure theory --- Distribution (Probability theory) --- Stochastic processes --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities
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The theory of large deviations deals with the evaluation, for a family of probability measures parameterized by a real valued variable, of the probabilities of events which decay exponentially in the parameter. Originally developed in the context of statistical mechanics and of (random) dynamical systems, it proved to be a powerful tool in the analysis of systems where the combined effects of random perturbations lead to a behavior significantly different from the noiseless case. The volume complements the central elements of this theory with selected applications in communication and control systems, bio-molecular sequence analysis, hypothesis testing problems in statistics, and the Gibbs conditioning principle in statistical mechanics. Starting with the definition of the large deviation principle (LDP), the authors provide an overview of large deviation theorems in ${{m I!R}}^d$ followed by their application. In a more abstract setup where the underlying variables take values in a topological space, the authors provide a collection of methods aimed at establishing the LDP, such as transformations of the LDP, relations between the LDP and Laplace's method for the evaluation for exponential integrals, properties of the LDP in topological vector spaces, and the behavior of the LDP under projective limits. They then turn to the study of the LDP for the sample paths of certain stochastic processes and the application of such LDP's to the problem of the exit of randomly perturbed solutions of differential equations from the domain of attraction of stable equilibria. They conclude with the LDP for the empirical measure of (discrete time) random processes: Sanov's theorem for the empirical measure of an i.i.d. sample, its extensions to Markov processes and mixing sequences and their application. The present soft cover edition is a corrected printing of the 1998 edition. Amir Dembo is a Professor of Mathematics and of Statistics at Stanford University. Ofer Zeitouni is a Professor of Mathematics at the Weizmann Institute of Science and at the University of Minnesota.
Electronic books. -- local. --- Large deviations. --- Limit theorems (Probability theory). --- Civil & Environmental Engineering --- Operations Research --- Engineering & Applied Sciences --- Limit theorems (Probability theory) --- Deviations, Large --- Mathematics. --- System theory. --- Probabilities. --- Systems Theory, Control. --- Probability Theory and Stochastic Processes. --- Probabilities --- Statistics --- Systems theory. --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Systems, Theory of --- Systems science --- Science --- Philosophy
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This thesis describes a method to control rare events in non-equilibrium systems by applying physical forces to those systems but without relying on numerical simulation techniques, such as copying rare events. In order to study this method, the book draws on the mathematical structure of equilibrium statistical mechanics, which connects large deviation functions with experimentally measureable thermodynamic functions. Referring to this specific structure as the “phenomenological structure for the large deviation principle”, the author subsequently extends it to time-series statistics that can be used to describe non-equilibrium physics. The book features pedagogical explanations and also shows many open problems to which the proposed method can be applied only to a limited extent. Beyond highlighting these challenging problems as a point of departure, it especially offers an effective means of description for rare events, which could become the next paradigm of non-equilibrium statistical mechanics.
Atomic Physics --- Physics --- Physical Sciences & Mathematics --- Time-series analysis. --- Large deviations. --- Deviations, Large --- Analysis of time series --- Limit theorems (Probability theory) --- Statistics --- Autocorrelation (Statistics) --- Harmonic analysis --- Mathematical statistics --- Probabilities --- Thermodynamics. --- Statistical physics. --- Complex Systems. --- Mathematical Physics. --- Statistical Physics and Dynamical Systems. --- Chemistry, Physical and theoretical --- Dynamics --- Mechanics --- Heat --- Heat-engines --- Quantum theory --- Statistical methods --- Dynamical systems. --- Mathematical physics. --- Physical mathematics --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Statics
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This Special Issue contains novel results in the area of out-of-equilibrium classical and quantum thermodynamics. Contributions are from different areas of physics, including statistical mechanics, quantum information and many-body systems.
quantum Otto engine --- Curzon–Ahlborn efficiency --- endoreversible quantum thermodynamics --- large deviations --- phase transitions --- condensation of fluctuations --- fluctuation relations --- magnetic cycle --- quantum otto cycle --- quantum thermodynamics --- quantum heat engines --- nonequilibrium systems --- ergotropy --- quantum correlations --- information thermodynamics --- collision model --- thermalization --- many-body quantum systems --- fluctuation relation --- Crooks equality --- coherence --- athermality --- photon added thermal state --- photon subtracted thermal state --- binomial states --- generalised coherent states --- laser cooling --- cavitation --- sonoluminescence --- fluctuation theorems --- collisional models
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The scope of the contributions to this book will be to present new and original research papers based on MPHIE, MHD, and MDPDE, as well as test statistics based on these estimators from a theoretical and applied point of view in different statistical problems with special emphasis on robustness. Manuscripts given solutions to different statistical problems as model selection criteria based on divergence measures or in statistics for high-dimensional data with divergence measures as loss function are considered. Reviews making emphasis in the most recent state-of-the art in relation to the solution of statistical problems base on divergence measures are also presented.
classification --- Bayes error rate --- Henze–Penrose divergence --- Friedman–Rafsky test statistic --- convergence rates --- bias and variance trade-off --- concentration bounds --- minimal spanning trees --- composite likelihood --- composite minimum density power divergence estimators --- model selection --- minimum pseudodistance estimation --- Robustness --- estimation of α --- monitoring --- numerical minimization --- S-estimation --- Tukey’s biweight --- integer-valued time series --- one-parameter exponential family --- minimum density power divergence estimator --- density power divergence --- robust change point test --- Galton-Watson branching processes with immigration --- Hellinger integrals --- power divergences --- Kullback-Leibler information distance/divergence --- relative entropy --- Renyi divergences --- epidemiology --- COVID-19 pandemic --- Bayesian decision making --- INARCH(1) model --- GLM model --- Bhattacharyya coefficient/distance --- time series of counts --- INGARCH model --- SPC --- CUSUM monitoring --- MDPDE --- contingency tables --- disparity --- mixed-scale data --- pearson residuals --- residual adjustment function --- robustness --- statistical distances --- Hellinger distance --- large deviations --- divergence measures --- rare event probabilities --- n/a --- Henze-Penrose divergence --- Friedman-Rafsky test statistic --- Tukey's biweight
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Data science, information theory, probability theory, statistical learning and other related disciplines greatly benefit from non-negative measures of dissimilarity between pairs of probability measures. These are known as divergence measures, and exploring their mathematical foundations and diverse applications is of significant interest. The present Special Issue, entitled “Divergence Measures: Mathematical Foundations and Applications in Information-Theoretic and Statistical Problems”, includes eight original contributions, and it is focused on the study of the mathematical properties and applications of classical and generalized divergence measures from an information-theoretic perspective. It mainly deals with two key generalizations of the relative entropy: namely, the R_ényi divergence and the important class of f -divergences. It is our hope that the readers will find interest in this Special Issue, which will stimulate further research in the study of the mathematical foundations and applications of divergence measures.
Bregman divergence --- f-divergence --- Jensen–Bregman divergence --- Jensen diversity --- Jensen–Shannon divergence --- capacitory discrimination --- Jensen–Shannon centroid --- mixture family --- information geometry --- difference of convex (DC) programming --- conditional Rényi divergence --- horse betting --- Kelly gambling --- Rényi divergence --- Rényi mutual information --- relative entropy --- chi-squared divergence --- f-divergences --- method of types --- large deviations --- strong data–processing inequalities --- information contraction --- maximal correlation --- Markov chains --- information inequalities --- mutual information --- Rényi entropy --- Carlson–Levin inequality --- information measures --- hypothesis testing --- total variation --- skew-divergence --- convexity --- Pinsker’s inequality --- Bayes risk --- statistical divergences --- minimum divergence estimator --- maximum likelihood --- bootstrap --- conditional limit theorem --- Bahadur efficiency --- α-mutual information --- Augustin–Csiszár mutual information --- data transmission --- error exponents --- dimensionality reduction --- discriminant analysis --- statistical inference --- n/a --- Jensen-Bregman divergence --- Jensen-Shannon divergence --- Jensen-Shannon centroid --- conditional Rényi divergence --- Rényi divergence --- Rényi mutual information --- strong data-processing inequalities --- Rényi entropy --- Carlson-Levin inequality --- Pinsker's inequality --- Augustin-Csiszár mutual information
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