Information Theory and the Central Limit Theorem

Information Theory and the Central Limit Theorem

Annotation. - Presents surprising, interesting connections between two apparently separate areas of mathematics- Written by one of the researchers who discovered these connections- Offers a new way of looking at familiar results.

Author: Oliver Johnson

Publisher: World Scientific

ISBN: 1860945376

Category: Computers

Page: 224

View: 238

Annotation. - Presents surprising, interesting connections between two apparently separate areas of mathematics- Written by one of the researchers who discovered these connections- Offers a new way of looking at familiar results.
Categories: Computers

A History of the Central Limit Theorem

A History of the Central Limit Theorem

In this context the book also describes the historical development of analytical probability theory and its tools, such as characteristic functions or moments.

Author: Hans Fischer

Publisher: Springer Science & Business Media

ISBN: 0387878572

Category: Mathematics

Page: 402

View: 851

This study discusses the history of the central limit theorem and related probabilistic limit theorems from about 1810 through 1950. In this context the book also describes the historical development of analytical probability theory and its tools, such as characteristic functions or moments. The central limit theorem was originally deduced by Laplace as a statement about approximations for the distributions of sums of independent random variables within the framework of classical probability, which focused upon specific problems and applications. Making this theorem an autonomous mathematical object was very important for the development of modern probability theory.
Categories: Mathematics

Stochastic Models Information Theory and Lie Groups Volume 1

Stochastic Models  Information Theory  and Lie Groups  Volume 1

Inf. Theory, 11, pp. 267I271, 1965. Braunstein, S.L., Caves, C.M., “Statistical
distance and the geometry of quantum states,” Phys. Rev. Lett., 72, pp.
3439I3443, 1994. Brown, L.D., “A proof of the Central Limit Theorem motivated by
the ...

Author: Gregory S. Chirikjian

Publisher: Springer Science & Business Media

ISBN: 9780817648039

Category: Mathematics

Page: 383

View: 619

This unique two-volume set presents the subjects of stochastic processes, information theory, and Lie groups in a unified setting, thereby building bridges between fields that are rarely studied by the same people. Unlike the many excellent formal treatments available for each of these subjects individually, the emphasis in both of these volumes is on the use of stochastic, geometric, and group-theoretic concepts in the modeling of physical phenomena. Stochastic Models, Information Theory, and Lie Groups will be of interest to advanced undergraduate and graduate students, researchers, and practitioners working in applied mathematics, the physical sciences, and engineering. Extensive exercises and motivating examples make the work suitable as a textbook for use in courses that emphasize applied stochastic processes or differential geometry.
Categories: Mathematics

Asymptotic Theory of Statistics and Probability

Asymptotic Theory of Statistics and Probability

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.

Author: Anirban DasGupta

Publisher: Springer Science & Business Media

ISBN: 9780387759708

Category: Mathematics

Page: 722

View: 242

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.
Categories: Mathematics

Classical and Quantum Information Theory

Classical and Quantum Information Theory

In the same limit, it can be shown that both distributions converge towards a
Gaussian (normal) PDF of same mean <k> and variance <12 I n61(1— q)Central-
limit theorem In probability theory, there exist several central-limit theorems,
which ...

Author: Emmanuel Desurvire

Publisher: Cambridge University Press

ISBN: 9780521881715

Category: Science

Page: 691

View: 566

This complete overview of classical and quantum information theory employs an informal yet accurate approach, for students, researchers and practitioners.
Categories: Science

Probability Theory

Probability Theory

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.

Author: Yakov G. Sinai

Publisher: Springer Science & Business Media

ISBN: 9783662028452

Category: Mathematics

Page: 140

View: 738

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.
Categories: Mathematics

Theory of U Statistics

Theory of U Statistics

The theory of U-statistics goes back to the fundamental work of Hoeffding [1], in which he proved the central limit theorem.

Author: Vladimir S. Korolyuk

Publisher: Springer Science & Business Media

ISBN: 9789401735155

Category: Mathematics

Page: 554

View: 610

The theory of U-statistics goes back to the fundamental work of Hoeffding [1], in which he proved the central limit theorem. During last forty years the interest to this class of random variables has been permanently increasing, and thus, the new intensively developing branch of probability theory has been formed. The U-statistics are one of the universal objects of the modem probability theory of summation. On the one hand, they are more complicated "algebraically" than sums of independent random variables and vectors, and on the other hand, they contain essential elements of dependence which display themselves in the martingale properties. In addition, the U -statistics as an object of mathematical statistics occupy one of the central places in statistical problems. The development of the theory of U-statistics is stipulated by the influence of the classical theory of summation of independent random variables: The law of large num bers, central limit theorem, invariance principle, and the law of the iterated logarithm we re proved, the estimates of convergence rate were obtained, etc.
Categories: Mathematics

Elements of Information Theory

Elements of Information Theory

We have also taken delight in relating Fisher information, mutual information, the
central limit theorem, and the Brunn–Minkowski and entropy power inequalities.
To our surprise, many of the classical results on determinant inequalities are ...

Author: Thomas M. Cover

Publisher: John Wiley & Sons

ISBN: 9781118585771

Category: Computers

Page: 776

View: 760

The latest edition of this classic is updated with new problem sets and material The Second Edition of this fundamental textbook maintains the book's tradition of clear, thought-provoking instruction. Readers are provided once again with an instructive mix of mathematics, physics, statistics, and information theory. All the essential topics in information theory are covered in detail, including entropy, data compression, channel capacity, rate distortion, network information theory, and hypothesis testing. The authors provide readers with a solid understanding of the underlying theory and applications. Problem sets and a telegraphic summary at the end of each chapter further assist readers. The historical notes that follow each chapter recap the main points. The Second Edition features: * Chapters reorganized to improve teaching * 200 new problems * New material on source coding, portfolio theory, and feedback capacity * Updated references Now current and enhanced, the Second Edition of Elements of Information Theory remains the ideal textbook for upper-level undergraduate and graduate courses in electrical engineering, statistics, and telecommunications.
Categories: Computers

An Introduction to Information Theory

An Introduction to Information Theory

... of information and probability theories. Chapter 7 gives a concise introduction
to multinormal distributions, laws of large numbers, and central-limit theorems.
These are essential tools for the proof of the main theorems of information theory.

Author: Fazlollah M. Reza

Publisher: Courier Corporation

ISBN: 9780486158440

Category: Mathematics

Page: 528

View: 605

Graduate-level study for engineering students presents elements of modern probability theory, information theory, coding theory, more. Emphasis on sample space, random variables, capacity, etc. Many reference tables and extensive bibliography. 1961 edition.
Categories: Mathematics

Principles and Practice of Information Theory

Principles and Practice of Information Theory

11 THE CENTRAL LIMIT THEOREM Information theory depends for its
development on the theorems of probability theory . To conclude this chapter , we
shall study a contribution of information theory back to probability theory .
Specifically ...

Author: Richard E. Blahut

Publisher: Addison Wesley Publishing Company

ISBN: UOM:39015065060108

Category: Mathematics

Page: 458

View: 901

Categories: Mathematics

Probability and Measure Theory

Probability and Measure Theory

Clear, readable style Solutions to many problems presented in text Solutions manual for instructors Material new to the second edition on ergodic theory, Brownian motion, and convergence theorems used in statistics No knowledge of general ...

Author: Robert B. Ash

Publisher: Academic Press

ISBN: 0120652021

Category: Mathematics

Page: 516

View: 213

Probability and Measure Theory, Second Edition, is a text for a graduate-level course in probability that includes essential background topics in analysis. It provides extensive coverage of conditional probability and expectation, strong laws of large numbers, martingale theory, the central limit theorem, ergodic theory, and Brownian motion. Clear, readable style Solutions to many problems presented in text Solutions manual for instructors Material new to the second edition on ergodic theory, Brownian motion, and convergence theorems used in statistics No knowledge of general topology required, just basic analysis and metric spaces Efficient organization
Categories: Mathematics

The Life and Times of the Central Limit Theorem

The Life and Times of the Central Limit Theorem

About the First Edition: The study of any topic becomes more meaningful if one also studies the historical development that resulted in the final theorem. ... This is an excellent book on mathematics in the making.

Author: William J. Adams

Publisher: American Mathematical Soc.

ISBN: 9780821848999

Category: Mathematics

Page: 195

View: 265

About the First Edition: The study of any topic becomes more meaningful if one also studies the historical development that resulted in the final theorem. ... This is an excellent book on mathematics in the making. --Philip Peak, The Mathematics Teacher, May, 1975 I find the book very interesting. It contains valuable information and useful references. It can be recommended not only to historians of science and mathematics but also to students of probability and statistics. --Wei-Ching Chang, Historica Mathematica, August, 1976 In the months since I wrote ... I have read it from cover to cover at least once and perused it here and there a number of times. I still find it a very interesting and worthwhile contribution to the history of probability and statistics. --Churchill Eisenhart, past president of the American Statistical Association, in a letter to the author, February 3, 1975 The name Central Limit Theorem covers a wide variety of results involving the determination of necessary and sufficient conditions under which sums of independent random variables, suitably standardized, have cumulative distribution functions close to the Gaussian distribution. As the name Central Limit Theorem suggests, it is a centerpiece of probability theory which also carries over to statistics. Part One of The Life and Times of the Central Limit Theorem, Second Edition traces its fascinating history from seeds sown by Jacob Bernoulli to use of integrals of $\exp (x^2)$ as an approximation tool, the development of the theory of errors of observation, problems in mathematical astronomy, the emergence of the hypothesis of elementary errors, the fundamental work of Laplace, and the emergence of an abstract Central Limit Theorem through the work of Chebyshev, Markov and Lyapunov. This closes the classical period of the life of the Central Limit Theorem, 1713-1901. The second part of the book includes papers by Feller and Le Cam, as well as comments by Doob, Trotter, and Pollard, describing the modern history of the Central Limit Theorem (1920-1937), in particular through contributions of Lindeberg, Cramer, Levy, and Feller. The Appendix to the book contains four fundamental papers by Lyapunov on the Central Limit Theorem, made available in English for the first time.
Categories: Mathematics

Uniform Central Limit Theorems

Uniform Central Limit Theorems

This expanded edition of the classic work on empirical processes now boasts several new proved theorems not in the first.

Author: R. M. Dudley

Publisher: Cambridge University Press

ISBN: 9780521498845

Category: Mathematics

Page: 486

View: 711

In this new edition of a classic work on empirical processes the author, an acknowledged expert, gives a thorough treatment of the subject with the addition of several proved theorems not included in the first edition, including the Bretagnolle–Massart theorem giving constants in the Komlos–Major–Tusnady rate of convergence for the classical empirical process, Massart's form of the Dvoretzky–Kiefer–Wolfowitz inequality with precise constant, Talagrand's generic chaining approach to boundedness of Gaussian processes, a characterization of uniform Glivenko–Cantelli classes of functions, Giné and Zinn's characterization of uniform Donsker classes, and the Bousquet–Koltchinskii–Panchenko theorem that the convex hull of a uniform Donsker class is uniform Donsker. The book will be an essential reference for mathematicians working in infinite-dimensional central limit theorems, mathematical statisticians, and computer scientists working in computer learning theory. Problems are included at the end of each chapter so the book can also be used as an advanced text.
Categories: Mathematics

Stochastic Models Information Theory and Lie Groups Volume 2

Stochastic Models  Information Theory  and Lie Groups  Volume 2

The main ideas to take away from this chapter are as follows: • It is possible to
describe SDEs and corresponding ... to uniformity, and in the case of noncompact
connected unimodular Lie groups, this results in a central limit theorem.

Author: Gregory S. Chirikjian

Publisher: Springer Science & Business Media

ISBN: 9780817649432

Category: Mathematics

Page: 435

View: 106

This two-volume set covers stochastic processes, information theory and Lie groups in a unified setting, bridging topics rarely studied together. The emphasis is on using stochastic, geometric, and group-theoretic concepts for modeling physical phenomena.
Categories: Mathematics

Martingale Limit Theory and Its Application

Martingale Limit Theory and Its Application

The text can be profitably used as a reference for mathematicians, advanced students, and professors of higher mathematics or statistics.

Author: P. Hall

Publisher: Academic Press

ISBN: 9781483263229

Category: Mathematics

Page: 320

View: 514

Martingale Limit Theory and Its Application discusses the asymptotic properties of martingales, particularly as regards key prototype of probabilistic behavior that has wide applications. The book explains the thesis that martingale theory is central to probability theory, and also examines the relationships between martingales and processes embeddable in or approximated by Brownian motion. The text reviews the martingale convergence theorem, the classical limit theory and analogs, and the martingale limit theorems viewed as the rate of convergence results in the martingale convergence theorem. The book explains the square function inequalities, weak law of large numbers, as well as the strong law of large numbers. The text discusses the reverse martingales, martingale tail sums, the invariance principles in the central limit theorem, and also the law of the iterated logarithm. The book investigates the limit theory for stationary processes via corresponding results for approximating martingales and the estimation of parameters from stochastic processes. The text can be profitably used as a reference for mathematicians, advanced students, and professors of higher mathematics or statistics.
Categories: Mathematics

1994 IEEE IMS Workshop on Information Theory and Statistics

1994 IEEE IMS Workshop on Information Theory and Statistics

These successes include the strong law of large numbers , the central limit
theorem , the law of the iterated logarithm , the ergodic theorem , and limit III .
REMARKS theorems for Markov processes . Certain theorems from information
theory ...

Author: IEEE Information Theory Society

Publisher: IEEE

ISBN: 0780327616

Category: Computers

Page: 107

View: 185

Categories: Computers

Probability Theory

Probability Theory

This clear exposition begins with basic concepts and moves on to combination of events, dependent events and random variables, Bernoulli trials and the De Moivre-Laplace theorem, and more. Includes 150 problems, many with answers.

Author: Y. A. Rozanov

Publisher: Courier Corporation

ISBN: 9780486321141

Category: Mathematics

Page: 148

View: 830

This clear exposition begins with basic concepts and moves on to combination of events, dependent events and random variables, Bernoulli trials and the De Moivre-Laplace theorem, and more. Includes 150 problems, many with answers.
Categories: Mathematics

An Introduction to the Theory of Point Processes

An Introduction to the Theory of Point Processes

This is the second volume of the reworked second edition of a key work on Point Process Theory.

Author: D.J. Daley

Publisher: Springer Science & Business Media

ISBN: 9780387213378

Category: Mathematics

Page: 573

View: 565

This is the second volume of the reworked second edition of a key work on Point Process Theory. Fully revised and updated by the authors who have reworked their 1988 first edition, it brings together the basic theory of random measures and point processes in a unified setting and continues with the more theoretical topics of the first edition: limit theorems, ergodic theory, Palm theory, and evolutionary behaviour via martingales and conditional intensity. The very substantial new material in this second volume includes expanded discussions of marked point processes, convergence to equilibrium, and the structure of spatial point processes.
Categories: Mathematics

Probability

Probability

Like its predecessor, this book starts from the premise that, rather than being a purely mathematical discipline, probability theory is an intimate companion of statistics.

Author: Allan Gut

Publisher: Springer Science & Business Media

ISBN: 9781461447078

Category: Mathematics

Page: 602

View: 920

This textbook on the theory of probability is aimed at graduate students. It starts with the basic tools, and goes on to cover a number of subjects in detail, including the three central planks of probability theory.
Categories: Mathematics