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Limit Theorems For Large Deviations

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Limit Theorems for Large Deviations

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Limit Theorems for Large Deviations by L. Saulis,V.A. Statulevicius Book Summary:

"Et moi ... - si j'avait su comment en revenir. One service mathematics has rendered the je n'y serais poin t aile.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O.H ea viside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non Iinearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service. topology has rendered mathematical physics .. .':: 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d 'e1:re of this series.

Uniform Limit Theorems for Sums of Independent Random Variables

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Uniform Limit Theorems for Sums of Independent Random Variables by Taĭvo Viktorovich Arak Book Summary:

Among the diverse constructions studied in modern probability theory, the scheme for summation of independent random variables occupies a special place. In the study of even this comparatively simple scheme it is possible to become familiar with the fundamental regularities characterizing the cumulative influence of a large number of random factors. Further, this abstract model is useful in many important practical situations. This book is devoted to the study of distributions of sums of independent random variables with minimal restrictions imposed on their distributions. The authors assume either that the distributions of the terms are concentrated on some finite interval to within a small mass, or that all the terms have the same, but arbitrary, distribution (or other conditions not connected with moment restrictions are introduced). Surprisingly, very substantive results are possible even under such a general statement of the problems.

Limit Theorems of Probability Theory

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Limit Theorems of Probability Theory by Yu.V. Prokhorov,V. Statulevicius Book Summary:

A collection of research level surveys on certain topics in probability theory by a well-known group of researchers. The book will be of interest to graduate students and researchers.

Refined Large Deviation Limit Theorems

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Refined Large Deviation Limit Theorems by Vladimir Vinogradov Book Summary:

This is a developing area of modern probability theory, which has applications in many areas. This volume is devoted to the systematic study of results on large deviations in situations where Cramér's condition on the finiteness of exponential moments may not be satisfied

Mod-φ Convergence

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Mod-φ Convergence by Valentin Féray,Pierre-Loïc Méliot,Ashkan Nikeghbali Book Summary:

The canonical way to establish the central limit theorem for i.i.d. random variables is to use characteristic functions and Lévy’s continuity theorem. This monograph focuses on this characteristic function approach and presents a renormalization theory called mod-φ convergence. This type of convergence is a relatively new concept with many deep ramifications, and has not previously been published in a single accessible volume. The authors construct an extremely flexible framework using this concept in order to study limit theorems and large deviations for a number of probabilistic models related to classical probability, combinatorics, non-commutative random variables, as well as geometric and number-theoretical objects. Intended for researchers in probability theory, the text is carefully well-written and well-structured, containing a great amount of detail and interesting examples.

Asymptotic Theory of Statistics and Probability

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Asymptotic Theory of Statistics and Probability by Anirban DasGupta Book Summary:

This unique book delivers an encyclopedic treatment of classic as well as contemporary large sample theory, dealing with both statistical problems and probabilistic issues and tools. The book is unique in its detailed coverage of fundamental topics. It is written in an extremely lucid style, with an emphasis on the conceptual discussion of the importance of a problem and the impact and relevance of the theorems. There is no other book in large sample theory that matches this book in coverage, exercises and examples, bibliography, and lucid conceptual discussion of issues and theorems.

Limit Theorems on Large Deviations for Markov Stochastic Processes

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Limit Theorems on Large Deviations for Markov Stochastic Processes by A.D. Wentzell Book Summary:

In recent decades a new branch of probability theory has been developing intensively, namely, limit theorems for stochastic processes. As compared to classical limit theorems for sums of independent random variables, the generalizations are going here in two directions simultaneously. First, instead of sums of independent variables one considers stochastic processes belonging to certain broad classes. Secondly, instead of the distribution of a single sum - the distribution of the value of a stochastic process at one (time) point - or the joint distribution of the values of a process at a finite number of points, one considers distributions in an infinite-dimensional function space. For stochastic processes constructed, starting from sums of independent random variables, this is the same as considering the joint distribution of an unboundedly increasing number of sums.

Hyperbolic Dynamics, Fluctuations and Large Deviations

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Hyperbolic Dynamics, Fluctuations and Large Deviations by D. Dolgopyat, Y. Pesin,M. Pollicott, L. Stoyanov Book Summary:

This volume contains the proceedings of the semester-long special program on Hyperbolic Dynamics, Large Deviations and Fluctuations, which was held from January-June 2013, at the Centre Interfacultaire Bernoulli, École Polytechnique Fédérale de Lausanne, Switzerland. The broad theme of the program was the long-term behavior of dynamical systems and their statistical behavior. During the last 50 years, the statistical properties of dynamical systems of many different types have been the subject of extensive study in statistical mechanics and thermodynamics, ergodic and probability theories, and some areas of mathematical physics. The results of this study have had a profound effect on many different areas in mathematics, physics, engineering and biology. The papers in this volume cover topics in large deviations and thermodynamics formalism and limit theorems for dynamic systems. The material presented is primarily directed at researchers and graduate students in the very broad area of dynamical systems and ergodic theory, but will also be of interest to researchers in related areas such as statistical physics, spectral theory and some aspects of number theory and geometry.

Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness

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Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness by Hubert Hennion,Loic Herve Book Summary:

The usefulness of from the of techniques perturbation theory operators, to kernel for limit theorems for a applied quasi-compact positive Q, obtaining Markov chains for stochastic of or dynamical by describing properties systems, of Perron- Frobenius has been demonstrated in several All use a operator, papers. these works share the features the features that must be same specific general ; used in each stem from the nature of the functional particular case precise space where the of is and from the number of quasi-compactness Q proved eigenvalues of of modulus 1. We here a functional framework for Q give general analytical this method and we the aforementioned behaviour within it. It asymptotic prove is worth that this framework is to allow the unified noticing sufficiently general treatment of all the cases considered in the literature the previously specific ; characters of model translate into the verification of of simple hypotheses every a functional nature. When to Markov kernels or to Perr- applied Lipschitz Frobenius associated with these statements rise operators expanding give maps, to new results and the of known The main clarify proofs already properties. of the deals with a Markov kernel for which 1 is a part quasi-compact Q paper of modulus 1. An essential but is not the simple eigenvalue unique eigenvalue element of the work is the of the of peripheral Q precise description spectrums and of its To conclude the the results obtained perturbations.

Probability Theory

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Probability Theory by Yakov G. Sinai Book Summary:

Sinai's book leads the student through the standard material for ProbabilityTheory, with stops along the way for interesting topics such as statistical mechanics, not usually included in a book for beginners. The first part of the book covers discrete random variables, using the same approach, basedon Kolmogorov's axioms for probability, used later for the general case. The text is divided into sixteen lectures, each covering a major topic. The introductory notions and classical results are included, of course: random variables, the central limit theorem, the law of large numbers, conditional probability, random walks, etc. Sinai's style is accessible and clear, with interesting examples to accompany new ideas. Besides statistical mechanics, other interesting, less common topics found in the book are: percolation, the concept of stability in the central limit theorem and the study of probability of large deviations. Little more than a standard undergraduate course in analysis is assumed of the reader. Notions from measure theory and Lebesgue integration are introduced in the second half of the text. The book is suitable for second or third year students in mathematics, physics or other natural sciences. It could also be usedby more advanced readers who want to learn the mathematics of probability theory and some of its applications in statistical physics.

Probability Theory

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Probability Theory by A A Borovkov Book Summary:

Probability theory forms the basis of mathematical statistics, and has applications in many related areas. This comprehensive book tackles the principal problems and advanced questions of probability theory in 21 self-contained chapters, which are presented in logical order, but are also easy to deal with individually. The book is further distinguished by the inclusion of clear and illustrative proofs of the fundamental results. Probability theory is currently an extremely active area of research internationally, and the importance of the Russian school in the development of the subject has long been recognized. The frequent references to Russian literature throughout this work lend a fresh dimension to the book, and make it an invaluable source of reference for Western researchers and advanced students in probability related subjects.

Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness

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Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness by Hubert Hennion,Loic Herve Book Summary:

This book shows how techniques from the perturbation theory of operators, applied to a quasi-compact positive kernel, may be used to obtain limit theorems for Markov chains or to describe stochastic properties of dynamical systems. A general framework for this method is given and then applied to treat several specific cases. An essential element of this work is the description of the peripheral spectra of a quasi-compact Markov kernel and of its Fourier-Laplace perturbations. This is first done in the ergodic but non-mixing case. This work is extended by the second author to the non-ergodic case. The only prerequisites for this book are a knowledge of the basic techniques of probability theory and of notions of elementary functional analysis.

Universal Theory for Strong Limit Theorems of Probability

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Universal Theory for Strong Limit Theorems of Probability by A. N. Frolov Book Summary:

This is the first book which the universal approach to strong laws of probability is discussed in. The universal theories are described for three important objects of probability theory: sums of independent random variables, processes with independent increments and renewal processes. Further generalizations are mentioned. Besides strong laws, large deviations are of independent interest. The case of infinite variations is considered as well. Readers can examine appropriate techniques and methods. Optimality of conditions is discussed.

Some Limit Theorems in Statistics

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Some Limit Theorems in Statistics by R. R. Bahadur Book Summary:

A discussion of topics in the theory of large deviations and of aspects of estimation and testing in large samples.

Large Deviations for Discrete-Time Processes with Averaging

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Large Deviations for Discrete-Time Processes with Averaging by O. V. Goulinskï,Alexander Yu Veretennikov Book Summary:

This book is mainly based on the Cramir--Chernoff renowned theorem, which deals with the 'rough' logarithmic asymptotics of the distribution of sums of independent, identically distributed random variables. The authors approach primarily the extensions of this theory to dependent, and in particular, nonmarkovian cases on function spaces. Recurrent algorithms of identification and adaptive control form the main examples behind the large deviation problems in this volume. The first part of the book exploits some ideas and concepts of the martingale approach, especially the concept of the stochastic exponential. The second part of the book covers Freindlin's approach, based on the Frobenius-type theorems for positive operators, which prove to be effective for the cases in consideration.

Large Deviations at Saint-Flour

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Large Deviations at Saint-Flour by Robert Azencott,Mark I. Freidlin,S.R.S. Varadhan Book Summary:

Contents: Azencott, R. : Large deviations and applications.- Freidlin, Mark I. Semi-linear PDE's and limit theorems for large deviations- Varadhan, Srinivasa R.S.: Large deviations and applications.

Probability Theory and Mathematical Statistics

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Probability Theory and Mathematical Statistics by B. Grigelionis Book Summary:

These Proceedings, published in two volumes contain the invited lectures and some selected short communications by international scientists which were presented at the Vth International Vilnius Conference on Probability Theory and Mathematical Statistics, Vilnius, Lithuania, 25 June--1 July 1989. Only available as a 2-volume set The main topics of presented papers cover the contempory techniques and results in limit theorems of probability and mathematical statistics, theory of Markov processes, martingales, random fields, stochastic physics, stochastic evolution equations, probability distributions on functional spaces and probabilistic number theory. In addition papers of leading specialists in quantum probability are included.

The Collected Works of Wassily Hoeffding

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The Collected Works of Wassily Hoeffding by Wassily Hoeffding Book Summary:

It has been a rare privilege to assemble this volume of Wassily Hoeffding's Collected Works. Wassily was, variously, a teacher, supervisor and colleague to us, and his work has had a profound influence on our own. Yet this would not be sufficient reason to publish his collected works. The additional and overwhelmingly compelling justification comes from the fun damental nature of his contributions to Statistics and Probability. Not only were his ideas original, and far-reaching in their implications; Wassily de veloped them so completely and elegantly in his papers that they are still cited as prime references up to half a century later. However, three of his earliest papers are cited rarely, if ever. These include material from his doctoral dissertation. They were written in German, and two of them were published in relatively obscure series. Rather than reprint the original articles, we have chosen to have them translated into English. These trans lations appear in this book, making Wassily's earliest research available to a wide audience for the first time. All other articles (including those of his contributions to Mathematical Reviews which go beyond a simple reporting of contents of articles) have been reproduced as they appeared, together with annotations and corrections made by Wassily on some private copies of his papers. Preceding these articles are three review papers which dis cuss the . impact of his work in some of the areas where he made major contributions.

Random Perturbations of Dynamical Systems

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Random Perturbations of Dynamical Systems by M. I. Freidlin,A. D. Wentzell Book Summary:

Asymptotical problems have always played an important role in probability theory. In classical probability theory dealing mainly with sequences of independent variables, theorems of the type of laws of large numbers, theorems of the type of the central limit theorem, and theorems on large deviations constitute a major part of all investigations. In recent years, when random processes have become the main subject of study, asymptotic investigations have continued to playa major role. We can say that in the theory of random processes such investigations play an even greater role than in classical probability theory, because it is apparently impossible to obtain simple exact formulas in problems connected with large classes of random processes. Asymptotical investigations in the theory of random processes include results of the types of both the laws of large numbers and the central limit theorem and, in the past decade, theorems on large deviations. Of course, all these problems have acquired new aspects and new interpretations in the theory of random processes.