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Large Deviations Techniques and Applications [Paperback]

Amir Dembo , Ofer Zeitouni

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Book Description

Nov. 18 2009 3642033105 978-3642033100 2nd ed. 1998. 2nd printing 2009

Large deviation estimates have proved to be the crucial tool required to handle many questions in statistics, engineering, statistial mechanics, and applied probability. Amir Dembo and Ofer Zeitouni, two of the leading researchers in the field, provide an introduction to the theory of large deviations and applications at a level suitable for graduate students. The mathematics is rigorous and the applications come from a wide range of areas, including electrical engineering and DNA sequences.

The second edition, printed in 1998, included new material on concentration inequalities and the metric and weak convergence approaches to large deviations. General statements and applications were sharpened, new exercises added, and the bibliography updated. The present soft cover edition is a corrected printing of the 1998 edition.

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From the Back Cover

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.

About the Author

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.

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Amazon.com: 4.0 out of 5 stars  2 reviews
4 of 4 people found the following review helpful
5.0 out of 5 stars Great book. Gives "all" the details. Requires some commitment. March 23 2011
By Ian Langmore - Published on Amazon.com
As part of postdoc work I wanted to study large deviations for solutions to PDE/ODE with random coefficients (not the usual additive stochastic noise). So I bought this book and read chapters 1, 2, 4, and parts of 3, 5, and 6).

This book provided "all" that I needed in order to obtain a simple result. In other words, after reading these chapters and doing some exercises one understands the common tricks and proof techniques needed. The chapters are well organized so one can for example read 1 and 2 and then understand the LDP for sums of independent random variables.

This book requires some commitment. For example, before getting into any interesting results, technical definitions such as "exponential tightness" are introduced. In contrast, Varadhan's lecture notes give you a general intuition and feeling for the important results right from the start (though I can't comment on how cleanly Varadhan fills in the details).

So buy this book not for an easy overview, but because you want to learn the techniques. Or maybe buy this book because you already know some techniques, in which case you could quickly scan the chapters and thus give yourself an overview.

P.S. The obligatory back-page comment on how some of the book can be read with "little more than basic calculus..." is (as always) completely false :)
2 of 6 people found the following review helpful
3.0 out of 5 stars A thorough treatment. But too difficult April 19 2010
By Yan Zhu - Published on Amazon.com
Maybe this is the only book which treats "Large deviation" so thoroughly and so rigorously. Many other books cite this one for further details. However, this book is very difficult to read. People, like me, with graduate level probability knowledge, will still feel difficult to follow. It is not all due to the presentation of this book. It may partially because the "theory of large deviation" itself is not in a good shape. So this book is not good one for self-study but might be a good reference.

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