Format:
Online-Ressource (XVI, 396p, digital)
ISBN:
9783642033117
Series Statement:
Stochastic Modelling and Applied Probability 38
Content:
LDP for Finite Dimensional Spaces -- Applications-The Finite Dimensional Case -- General Principles -- Sample Path Large Deviations -- The LDP for Abstract Empirical Measures -- Applications of Empirical Measures LDP.
Content:
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 ${{\rm I\!R}}^d$ 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.
Note:
Includes bibliographical references and index
,
""Preface to the Second Edition""; ""Preface to the First Edition""; ""Contents""; ""Introduction""; ""Rare Events and Large Deviations""; ""The Large Deviation Principle""; ""Historical Notes and References""; ""LDP for Finite Dimensional Spaces""; ""Combinatorial Techniques for Finite Alphabets""; ""The Method of Types and Sanov's Theorem""; ""Cramér's Theorem for Finite Alphabets in I-R""; ""Large Deviations for Sampling Without Replacement""; ""Cramér's Theorem""; ""Cramér's Theorem in I-R""; ""Cramér's Theorem in I-Rd""; ""The G�rtner-Ellis Theorem""; ""Concentration Inequalities""
,
""Inequalities for Bounded Martingale Differences""""Talagrand's Concentration Inequalities""; ""Historical Notes and References""; ""Applications - The Finite Dimensional Case""; ""Large Deviations for Finite State Markov Chains""; ""LDP for Additive Functionals of Markov Chains""; ""Sanov's Theorem for the Empirical Measure of Markov Chains""; ""Sanov's Theorem for the Pair Empirical Measure of Markov Chains""; ""Long Rare Segments in Random Walks""; ""The Gibbs Conditioning Principle for Finite Alphabets""; ""The Hypothesis Testing Problem""
,
""Generalized Likelihood Ratio Test for Finite Alphabets""""Rate Distortion Theory ""; ""Moderate Deviations and Exact Asymptotics in I-Rd""; ""Historical Notes and References""; ""General Principles""; ""Existence of an LDP and Related Properties""; ""Properties of the LDP ""; ""The Existence of an LDP ""; ""Transformations of LDPs""; ""Contraction Principles""; ""Exponential Approximations""; ""Varadhan's Integral Lemma""; ""Bryc's Inverse Varadhan Lemma""; ""LDP in Topological Vector Spaces""; ""A General Upper Bound""; ""Convexity Considerations""; ""Abstract GÃ?rtner-Ellis Theorem""
,
""Large Deviations for Projective Limits""""The LDP and Weak Convergence in Metric Spaces""; ""Historical Notes and References""; ""Sample Path Large Deviations""; ""Sample Path Large Deviations for Random Walks""; ""Brownian Motion Sample Path Large Deviations""; ""Multivariate Random Walk and Brownian Sheet""; ""Performance Analysis of DMPSK Modulation""; ""Large Exceedances in I-Rd""; ""The Freidlin-Wentzell Theory""; ""The Problem of Diffusion Exit from a Domain""; ""The Performance of Tracking Loops""; ""An Angular Tracking Loop Analysis""; ""The Analysis of Range Tracking Loops""
,
""Historical Notes and References""""The LDP for Abstract Empirical Measures""; ""Cramér's Theorem in Polish Spaces""; ""Sanov's Theorem""; ""LDP for the Empirical Measure - The Uniform Markov Case""; ""Mixing Conditions and LDP ""; ""LDP for the Empirical Mean in I-Rd""; ""Empirical Measure LDP for Mixing Processes""; ""LDP for Empirical Measures of Markov Chains""; ""LDP for Occupation Times""; ""LDP for the k-Empirical Measures""; ""Process Level LDP for Markov Chains""; ""A Weak Convergence Approach to Large Deviations""; ""Historical Notes and References""
,
""Applications of Empirical Measures LDP ""
Additional Edition:
ISBN 9783642033100
Additional Edition:
Buchausg. u.d.T. Dembo, Amir Large deviations techniques and applications Berlin : Springer, 2010 ISBN 9783642033100
Additional Edition:
Erscheint auch als Druck-Ausgabe Dembo, Amir Large deviations techniques and applications New York : Springer, 1998 ISBN 0387984062
Language:
English
Subjects:
Mathematics
Keywords:
Große Abweichung
;
Große Abweichung
DOI:
10.1007/978-3-642-03311-7
URL:
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