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This volume is devoted to the study of asymptotic properties of wide classes of stochastic systems arising in mathematical statistics, percolation theory, statistical physics and reliability theory. Attention is paid not only to positive and negative associations introduced in the pioneering papers by Harris, Lehmann, Esary, Proschan, Walkup, Fortuin, Kasteleyn and Ginibre, but also to new and more general dependence conditions. Naturally, this scope comprises families of independent real-valued random variables. A variety of important results and examples of Markov processes, random measures

Random fields. --- Limit theorems (Probability theory) --- Probabilities --- Fields, Random --- Stochastic processes --- Random fields --- Champs aléatoires --- Théorèmes limites (Théorie des probabilités)

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Randomized algorithms have become a central part of the algorithms curriculum, based on their increasingly widespread use in modern applications. This book presents a coherent and unified treatment of probabilistic techniques for obtaining high probability estimates on the performance of randomized algorithms. It covers the basic toolkit from the Chernoff-Hoeffding bounds to more sophisticated techniques like martingales and isoperimetric inequalities, as well as some recent developments like Talagrand's inequality, transportation cost inequalities and log-Sobolev inequalities. Along the way, variations on the basic theme are examined, such as Chernoff-Hoeffding bounds in dependent settings. The authors emphasise comparative study of the different methods, highlighting respective strengths and weaknesses in concrete example applications. The exposition is tailored to discrete settings sufficient for the analysis of algorithms, avoiding unnecessary measure-theoretic details, thus making the book accessible to computer scientists as well as probabilists and discrete mathematicians.

Random variables. --- Distribution (Probability theory) --- Limit theorems (Probability theory) --- Algorithms. --- Algorism --- Algebra --- Arithmetic --- Probabilities --- Distribution functions --- Frequency distribution --- Characteristic functions --- Chance variables --- Stochastic variables --- Variables (Mathematics) --- Foundations

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"The theory of empirical processes provides tools for the development of asymptotic theory in (non-parametric) statistical models, and possibly the unified treatment of a number of them. This book reveals the relation between the asymptotic behaviour of M-estimators and the complexity of parameter space. Virtually all results are proved using only elementary ideas developed within the book; there is minimal recourse to abstract theoretical results. To make the results concrete, a detailed treatment is presented for two important examples of M-estimation, namely maximum likelihood and least squares. The theory also covers estimation methods using penalties and sieves." "Many illustrative examples are given, including the Grenander estimator, estimation of functions of bounded variation, smoothing splines, partially linear models, mixture models and image analysis." "For Graduate students and professionals in statistics, as well as those with an interest in applications to such areas as econometrics, medical statistics, etc."--Jacket.

Nonparametric statistics. --- Estimation theory. --- Limit theorems (Probability theory) --- mathematische modellen, toegepast op economie --- waarschijnlijkheidsleer --- wiskundige statistiek --- Probabilities --- Estimating techniques --- Least squares --- Mathematical statistics --- Stochastic processes --- Distribution-free statistics --- Statistics, Distribution-free --- Statistics, Nonparametric --- Nonparametric statistics --- Estimation theory --- Limit theorems (Probability theory).

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This book provides a comprehensive description of a new method of proving the central limit theorem, through the use of apparently unrelated results from information theory. It gives a basic introduction to the concepts of entropy and Fisher information, and collects together standard results concerning their behaviour. It brings together results from a number of research papers as well as unpublished material, showing how the techniques can give a unified view of limit theorems.

Central limit theorem. --- Information theory --- Probabilities. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Communication theory --- Communication --- Cybernetics --- Asymptotic distribution (Probability theory) --- Limit theorems (Probability theory) --- Statistical methods.

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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