## Probability: The Classical Limit Theorems

**Author**: Henry McKean

**Publisher:**Cambridge University Press

**ISBN:**1107053218

**Category:**Computers

**Page:**488

**View:**2835

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A leading authority sheds light on a variety of interesting topics in which probability theory plays a key role.

## A History of the Central Limit Theorem

*From Classical to Modern Probability Theory*

**Author**: Hans Fischer

**Publisher:**Springer Science & Business Media

**ISBN:**9780387878577

**Category:**Mathematics

**Page:**402

**View:**9774

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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.

## Limit Theorems of Probability Theory

**Author**: Yu.V. Prokhorov,V. Statulevicius

**Publisher:**Springer Science & Business Media

**ISBN:**3662041723

**Category:**Mathematics

**Page:**273

**View:**2288

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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.

## Probability

**Author**: Davar Khoshnevisan

**Publisher:**American Mathematical Soc.

**ISBN:**0821842153

**Category:**Mathematics

**Page:**224

**View:**9820

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This is a textbook for a one-semester graduate course in measure-theoretic probability theory, but with ample material to cover an ordinary year-long course at a more leisurely pace. Khoshnevisan's approach is to develop the ideas that are absolutely central to modern probability theory, and to showcase them by presenting their various applications. As a result, a few of the familiar topics are replaced by interesting non-standard ones. The topics range from undergraduate probability and classical limit theorems to Brownian motion and elements of stochastic calculus. Throughout, the reader will find many exciting applications of probability theory and probabilistic reasoning. There are numerous exercises, ranging from the routine to the very difficult. Each chapter concludes with historical notes.

## Theory of Probability

**Author**: Boris V. Gnedenko

**Publisher:**CRC Press

**ISBN:**9789056995850

**Category:**Mathematics

**Page:**520

**View:**2142

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This book is the sixth edition of a classic text that was first published in 1950 in the former Soviet Union. The clear presentation of the subject and extensive applications supported with real data helped establish the book as a standard for the field. To date, it has been published into more that ten languages and has gone through five editions. The sixth edition is a major revision over the fifth. It contains new material and results on the Local Limit Theorem, the Integral Law of Large Numbers, and Characteristic Functions. The new edition retains the feature of developing the subject from intuitive concepts and demonstrating techniques and theory through large numbers of examples. The author has, for the first time, included a brief history of probability and its development. Exercise problems and examples have been revised and new ones added.

## Probability

*The Classical Limit Theorems*

**Author**: Henry McKean

**Publisher:**Cambridge University Press

**ISBN:**131606249X

**Category:**Mathematics

**Page:**N.A

**View:**8741

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Probability theory has been extraordinarily successful at describing a variety of phenomena, from the behaviour of gases to the transmission of messages, and is, besides, a powerful tool with applications throughout mathematics. At its heart are a number of concepts familiar in one guise or another to many: Gauss' bell-shaped curve, the law of averages, and so on, concepts that crop up in so many settings they are in some sense universal. This universality is predicted by probability theory to a remarkable degree. This book explains that theory and investigates its ramifications. Assuming a good working knowledge of basic analysis, real and complex, the author maps out a route from basic probability, via random walks, Brownian motion, the law of large numbers and the central limit theorem, to aspects of ergodic theorems, equilibrium and nonequilibrium statistical mechanics, communication over a noisy channel, and random matrices. Numerous examples and exercises enrich the text.

## The Life and Times of the Central Limit Theorem

**Author**: William J. Adams

**Publisher:**American Mathematical Soc.

**ISBN:**0821848992

**Category:**Mathematics

**Page:**195

**View:**1824

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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.

## Wahrscheinlichkeitstheorie und Stochastische Prozesse

**Author**: Michael Mürmann

**Publisher:**Springer-Verlag

**ISBN:**364238160X

**Category:**Mathematics

**Page:**428

**View:**861

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Dieses Lehrbuch beschäftigt sich mit den zentralen Gebieten einer maßtheoretisch orientierten Wahrscheinlichkeitstheorie im Umfang einer zweisemestrigen Vorlesung. Nach den Grundlagen werden Grenzwertsätze und schwache Konvergenz behandelt. Es folgt die Darstellung und Betrachtung der stochastischen Abhängigkeit durch die bedingte Erwartung, die mit der Radon-Nikodym-Ableitung realisiert wird. Sie wird angewandt auf die Theorie der stochastischen Prozesse, die nach der allgemeinen Konstruktion aus der Untersuchung von Martingalen und Markov-Prozessen besteht. Neu in einem Lehrbuch über allgemeine Wahrscheinlichkeitstheorie ist eine Einführung in die stochastische Analysis von Semimartingalen auf der Grundlage einer geeigneten Stetigkeitsbedingung mit Anwendungen auf die Theorie der Finanzmärkte. Das Buch enthält zahlreiche Übungen, teilweise mit Lösungen. Neben der Theorie vertiefen Anmerkungen, besonders zu mathematischen Modellen für Phänomene der Realität, das Verständnis.

## Stable Convergence and Stable Limit Theorems

**Author**: Erich Häusler,Harald Luschgy

**Publisher:**Springer

**ISBN:**331918329X

**Category:**Mathematics

**Page:**228

**View:**5131

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The authors present a concise but complete exposition of the mathematical theory of stable convergence and give various applications in different areas of probability theory and mathematical statistics to illustrate the usefulness of this concept. Stable convergence holds in many limit theorems of probability theory and statistics – such as the classical central limit theorem – which are usually formulated in terms of convergence in distribution. Originated by Alfred Rényi, the notion of stable convergence is stronger than the classical weak convergence of probability measures. A variety of methods is described which can be used to establish this stronger stable convergence in many limit theorems which were originally formulated only in terms of weak convergence. Naturally, these stronger limit theorems have new and stronger consequences which should not be missed by neglecting the notion of stable convergence. The presentation will be accessible to researchers and advanced students at the master's level with a solid knowledge of measure theoretic probability.

## Probability Theory and Mathematical Statistics

**Author**: Ibragimoc

**Publisher:**CRC Press

**ISBN:**9782919875146

**Category:**Mathematics

**Page:**320

**View:**5413

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The topics treated fall into three main groups, all of which deal with classical problems which originated in the work of Kolmogorov. The first section looks at probability limit theorems, the second deals with stochastic analysis, and the final part presents some papers on non-parametric and semi-parametric models of mathematical statistics and asymptotic problems. The contributions come from some of the foremost mathematicians in the world today, making for a truly international collection of papers, permeated with the influence of Kolmogorov's works.

## Modern Theory of Summation of Random Variables

**Author**: Vladimir M. Zolotarev

**Publisher:**Walter de Gruyter

**ISBN:**3110936534

**Category:**Mathematics

**Page:**426

**View:**3868

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The series is devoted to the publication of high-level monographs and surveys which cover the whole spectrum of probability and statistics. The books of the series are addressed to both experts and advanced students.

## Probabilities on the Heisenberg Group

*Limit Theorems and Brownian Motion*

**Author**: Daniel Neuenschwander

**Publisher:**Springer

**ISBN:**3540685901

**Category:**Mathematics

**Page:**148

**View:**2911

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The Heisenberg group comes from quantum mechanics and is the simplest non-commutative Lie group. While it belongs to the class of simply connected nilpotent Lie groups, it turns out that its special structure yields many results which (up to now) have not carried over to this larger class. This book is a survey of probabilistic results on the Heisenberg group. The emphasis lies on limit theorems and their relation to Brownian motion. Besides classical probability tools, non-commutative Fourier analysis and functional analysis (operator semigroups) comes in. The book is intended for probabilists and analysts interested in Lie groups, but given the many applications of the Heisenberg group, it will also be useful for theoretical phycisists specialized in quantum mechanics and for engineers.

## Limit Theorems for Multi-Indexed Sums of Random Variables

**Author**: Oleg Klesov

**Publisher:**Springer

**ISBN:**3662443880

**Category:**Mathematics

**Page:**483

**View:**4911

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Presenting the first unified treatment of limit theorems for multiple sums of independent random variables, this volume fills an important gap in the field. Several new results are introduced, even in the classical setting, as well as some new approaches that are simpler than those already established in the literature. In particular, new proofs of the strong law of large numbers and the Hajek-Renyi inequality are detailed. Applications of the described theory include Gibbs fields, spin glasses, polymer models, image analysis and random shapes. Limit theorems form the backbone of probability theory and statistical theory alike. The theory of multiple sums of random variables is a direct generalization of the classical study of limit theorems, whose importance and wide application in science is unquestionable. However, to date, the subject of multiple sums has only been treated in journals. The results described in this book will be of interest to advanced undergraduates, graduate students and researchers who work on limit theorems in probability theory, the statistical analysis of random fields, as well as in the field of random sets or stochastic geometry. The central topic is also important for statistical theory, developing statistical inferences for random fields, and also has applications to the sciences, including physics and chemistry.

## Probability Theory III

*Stochastic Calculus*

**Author**: Yurij V. Prokhorov,Albert N. Shiryaev

**Publisher:**Springer Science & Business Media

**ISBN:**3662036401

**Category:**Mathematics

**Page:**256

**View:**6687

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This volume of the Encyclopaedia is a survey of stochastic calculus, an increasingly important part of probability, authored by well-known experts in the field. The book addresses graduate students and researchers in probability theory and mathematical statistics, as well as physicists and engineers who need to apply stochastic methods.

## A Natural Introduction to Probability Theory

**Author**: Ronald Meester

**Publisher:**Springer Science & Business Media

**ISBN:**9783764321888

**Category:**Mathematics

**Page:**191

**View:**3296

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"The book provides an introduction, in full rigour, of discrete and continuous probability, without using algebras or sigma-algebras; only familiarity with first-year calculus is required. Starting with the framework of discrete probability, it is already possible to discuss random walk, weak laws of large numbers and a first central limit theorem. After that, continuous probability, infinitely many repetitions, strong laws of large numbers, and branching processes are extensively treated. Finally, weak convergence is introduced and the central limit theorem is proved." "The theory is illustrated with many original and surprising examples and problems, taken from classical applications like gambling, geometry or graph theory, as well as from applications in biology, medicine, social sciences, sports, and coding theory."--BOOK JACKET.

## Lectures in Probability Theory and Mathematical Statistics

**Author**: Stefan Zubrzycki

**Publisher:**Elsevier Publishing Company

**ISBN:**N.A

**Category:**Mathematical statistics

**Page:**321

**View:**3673

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Events and probability; Probabilities in the most important spaces of elementary events; Conditional probability, stochastic independence of events; Classical limit theorems; Random variables; Laws of large numbers; Central limit theorems of probability theory; Statistical inference.

## Strong Limit Theorems in Noncommutative L2-Spaces

**Author**: Ryszard Jajte

**Publisher:**Springer

**ISBN:**9783540542148

**Category:**Mathematics

**Page:**113

**View:**7023

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The noncommutative versions of fundamental classical results on the almost sure convergence in L2-spaces are discussed: individual ergodic theorems, strong laws of large numbers, theorems on convergence of orthogonal series, of martingales of powers of contractions etc. The proofs introduce new techniques in von Neumann algebras. The reader is assumed to master the fundamentals of functional analysis and probability. The book is written mainly for mathematicians and physicists familiar with probability theory and interested in applications of operator algebras to quantum statistical mechanics.

## Probability and Stochastics

**Author**: Erhan Çınlar

**Publisher:**Springer Science & Business Media

**ISBN:**9780387878591

**Category:**Mathematics

**Page:**558

**View:**1023

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This text is an introduction to the modern theory and applications of probability and stochastics. The style and coverage is geared towards the theory of stochastic processes, but with some attention to the applications. In many instances the gist of the problem is introduced in practical, everyday language and then is made precise in mathematical form. The first four chapters are on probability theory: measure and integration, probability spaces, conditional expectations, and the classical limit theorems. There follows chapters on martingales, Poisson random measures, Levy Processes, Brownian motion, and Markov Processes. Special attention is paid to Poisson random measures and their roles in regulating the excursions of Brownian motion and the jumps of Levy and Markov processes. Each chapter has a large number of varied examples and exercises. The book is based on the author’s lecture notes in courses offered over the years at Princeton University. These courses attracted graduate students from engineering, economics, physics, computer sciences, and mathematics. Erhan Cinlar has received many awards for excellence in teaching, including the President’s Award for Distinguished Teaching at Princeton University. His research interests include theories of Markov processes, point processes, stochastic calculus, and stochastic flows. The book is full of insights and observations that only a lifetime researcher in probability can have, all told in a lucid yet precise style.

## Lectures on Probability Theory and Statistics

*Ecole d'Eté de Probabilités de Saint-Flour XXXIII - 2003*

**Author**: Amir Dembo,Tadahisa Funaki

**Publisher:**Springer

**ISBN:**3540315373

**Category:**Mathematics

**Page:**286

**View:**5545

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This volume contains two of the three lectures that were given at the 33rd Probability Summer School in Saint-Flour (July 6-23, 2003). Amir Dembo’s course is devoted to recent studies of the fractal nature of random sets, focusing on some fine properties of the sample path of random walk and Brownian motion. In particular, the cover time for Markov chains, the dimension of discrete limsup random fractals, the multi-scale truncated second moment and the Ciesielski-Taylor identities are explored. Tadahisa Funaki’s course reviews recent developments of the mathematical theory on stochastic interface models, mostly on the so-called \nabla \varphi interface model. The results are formulated as classical limit theorems in probability theory, and the text serves with good applications of basic probability techniques.