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Probability theory --- Large deviations --- Grandes déviations --- Deviations, Large --- Limit theorems (Probability theory) --- Statistics --- Grandes déviations
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Mathematical statistics --- Large deviations --- Grandes déviations --- Deviations, Large --- Limit theorems (Probability theory) --- Statistics --- Large deviations. --- Grandes déviations
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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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Large deviations --- Limit theorems (Probability theory) --- Phase transformations (Statistical physics) --- Tunneling (Physics) --- Penetration probability --- Quantum mechanical tunneling --- Tunnel effect --- Electric conductivity --- Solids --- Phase changes (Statistical physics) --- Phase transitions (Statistical physics) --- Phase rule and equilibrium --- Statistical physics --- Probabilities --- Deviations, Large --- Statistics --- Effet tunnel --- Transitions de phases --- Théorèmes des limites (théorie des probabilités) --- Grandes déviations --- Effet tunnel. --- Transitions de phases. --- Grandes déviations.
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Large deviations. --- Markov processes. --- Additive functions. --- Grandes déviations --- Markov, Processus de --- Fonctions additives --- Stochastic processes --- Additive functions --- Large deviations --- Markov processes --- 51 <082.1> --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Deviations, Large --- Limit theorems (Probability theory) --- Statistics --- Functions --- Mathematics--Series
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Ergodic theory. Information theory --- Central limit theorem --- Large deviations --- Markov processes --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Stochastic processes --- Deviations, Large --- Limit theorems (Probability theory) --- Statistics --- Asymptotic distribution (Probability theory) --- Central limit theorem. --- Théorème de la limite centrale --- Markov processes. --- Markov, Processus de --- Large deviations. --- Grandes déviations --- Théorème de la limite centrale. --- Markov, Processus de. --- Grandes déviations.
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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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