## Probability: The Classical Limit Theorems

**Author**: Henry McKean

**Publisher:**Cambridge University Press

**ISBN:**1107053218

**Category:**Computers

**Page:**488

**View:**4304

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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:**5709

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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.

## Mathematik

**Author**: Timothy Gowers

**Publisher:**N.A

**ISBN:**9783150187067

**Category:**

**Page:**207

**View:**5969

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## Limit Theorems of Probability Theory

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

**Publisher:**Springer Science & Business Media

**ISBN:**3662041723

**Category:**Mathematics

**Page:**273

**View:**8356

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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.

## Wahrscheinlichkeitstheorie und Stochastische Prozesse

**Author**: Michael Mürmann

**Publisher:**Springer-Verlag

**ISBN:**364238160X

**Category:**Mathematics

**Page:**428

**View:**6387

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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.

## Probability

*The Classical Limit Theorems*

**Author**: Henry McKean

**Publisher:**Cambridge University Press

**ISBN:**131606249X

**Category:**Mathematics

**Page:**N.A

**View:**1168

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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:**3260

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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.

## Grundz?ge einer allgemeinen theorie der linearen integralgleichungen

**Author**: D. Hilbert

**Publisher:**Рипол Классик

**ISBN:**5880918734

**Category:**History

**Page:**230

**View:**3636

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## Theory of Probability

**Author**: Boris V. Gnedenko

**Publisher:**CRC Press

**ISBN:**9789056995850

**Category:**Mathematics

**Page:**520

**View:**4284

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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.

## Stable Convergence and Stable Limit Theorems

**Author**: Erich Häusler,Harald Luschgy

**Publisher:**Springer

**ISBN:**331918329X

**Category:**Mathematics

**Page:**228

**View:**1907

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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.

## Uniform Central Limit Theorems

**Author**: R. M. Dudley

**Publisher:**Cambridge University Press

**ISBN:**0521498848

**Category:**Mathematics

**Page:**486

**View:**6793

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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.

## Probability

**Author**: Davar Khoshnevisan

**Publisher:**American Mathematical Soc.

**ISBN:**0821842153

**Category:**Mathematics

**Page:**224

**View:**6169

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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.

## Limit Theory for Mixing Dependent Random Variables

**Author**: Lin Zhengyan,Lu Chuanrong

**Publisher:**Springer Science & Business Media

**ISBN:**9780792342199

**Category:**Mathematics

**Page:**430

**View:**7365

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For many practical problems, observations are not independent. In this book, limit behaviour of an important kind of dependent random variables, the so-called mixing random variables, is studied. Many profound results are given, which cover recent developments in this subject, such as basic properties of mixing variables, powerful probability and moment inequalities, weak convergence and strong convergence (approximation), limit behaviour of some statistics with a mixing sample, and many useful tools are provided. Audience: This volume will be of interest to researchers and graduate students in the field of probability and statistics, whose work involves dependent data (variables).

## Modern Theory of Summation of Random Variables

**Author**: Vladimir M. Zolotarev

**Publisher:**Walter de Gruyter

**ISBN:**3110936534

**Category:**Mathematics

**Page:**426

**View:**3862

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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.

## Lectures on Probability Theory and Statistics

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

**Author**: Amir Dembo,Tadahisa Funaki

**Publisher:**Springer Science & Business Media

**ISBN:**9783540260691

**Category:**Mathematics

**Page:**281

**View:**6712

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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.

## Asymptotische Gesetƶe der Wahrscheinlichkeitsrechnung

**Author**: A. Khintchine

**Publisher:**Springer

**ISBN:**9783642494604

**Category:**Computers

**Page:**79

**View:**3452

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Das sachliche Hauptziel der Wahrscheinlichkeitsrechnung ist die mathematische Erforschung von Massenerscheinungen. In formaler Hin sicht bedeutet das einen erkenntnistheoretisch genügend scharf ab gegrenzten Problemkreis: diejenigen Gesetzmäßigkeiten der Erscheinun gen und Vorgänge theoretisch zu erfassen, die durch das Massenhafte an ihnen (d. h. durch das Auftreten einer großen Anzahl von in gewissem Sinne gleichberechtigten Ereignissen, Größen u. dgl. m. ) in ihren Haupt zügen bedingt sind, so daß daneben die individuelle Beschaffenheit der einzelnen Ingredienten gewissermaßen in den Hintergrund tritt. Rein mathematisch führt das endlich zu Infinitesimalbetrachtungen einer spezifischen Gattung, indem die für eine unendlich große Ingredienten anzahl geltenden Grenzgesetze systematisch untersucht und begründet werden. In diesem Zusammenhang erscheinen die unter dem Namen von "Grenzwertsätzen" bekannten asymptotischen Gesetze der Wahr scheinlichkeitsrechnung keinesfalls als ein isoliertes Nebenstück dieser Wissenschaft, sondern sie bilden im Gegenteil den wesentlichsten Teil ihrer Problematik. Diese "asymptotische" Wahrscheinlichkeitsrechnung ist als mathe matische Wissenschaft noch ziemlich weit davon entfernt, ein einheit liches Ganzes zu bilden. Vor wenigen Jahren zählte sie zu ihren Ergeb nissen nur ein paar ganz abgesondert stehender, durch keinen allgemeinen Standpunkt vereinigter Grenzwertsätze. Nur in der allerletzten Zeit konnte sie gewisse neue Aussichtspunkte erringen, die die Hoffnung erwecken, für dieses theoretisch grundlegende und auch für die Natur wissenschaften äußerst wichtige Forschungsgebiet in absehbarer Zeit eine einheitliche Theorie zu gewinnen. Es müssen hier einerseits die aus der physikalischen Statistik kommenden, mit der sog.

## Probability Theory

*Philosophy, Recent History and Relations to Science*

**Author**: Vincent F. Hendricks,Stig Andur Pedersen,Klaus Frovin Jørgensen

**Publisher:**Springer Science & Business Media

**ISBN:**9780792369523

**Category:**Mathematics

**Page:**183

**View:**2030

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A collection of papers presented at the conference on Probability Theory - Philosophy, Recent History and Relations to Science, University of Roskilde, Denmark, September 16-18, 1998. Since the measure theoretical definition of probability was proposed by Kolmogorov, probability theory has developed into a mature mathematical theory. It is today a fruitful field of mathematics that has important applications in philosophy, science, engineering, and many other areas. The measure theoretical definition of probability and its axioms, however, are not without their problems; some of them even puzzled Kolmogorov. This book sheds light on some recent discussions of the problems in probability theory and their history, analysing their philosophical and mathematical significance, and the role pf mathematical probability theory in other sciences.

## Brownian Motion

*An Introduction to Stochastic Processes*

**Author**: René L. Schilling,Lothar Partzsch

**Publisher:**Walter de Gruyter GmbH & Co KG

**ISBN:**3110307308

**Category:**Mathematics

**Page:**424

**View:**8325

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Brownian motion is one of the most important stochastic processes in continuous time and with continuous state space. Within the realm of stochastic processes, Brownian motion is at the intersection of Gaussian processes, martingales, Markov processes, diffusions and random fractals, and it has influenced the study of these topics. Its central position within mathematics is matched by numerous applications in science, engineering and mathematical finance. Often textbooks on probability theory cover, if at all, Brownian motion only briefly. On the other hand, there is a considerable gap to more specialized texts on Brownian motion which is not so easy to overcome for the novice. The authors’ aim was to write a book which can be used as an introduction to Brownian motion and stochastic calculus, and as a first course in continuous-time and continuous-state Markov processes. They also wanted to have a text which would be both a readily accessible mathematical back-up for contemporary applications (such as mathematical finance) and a foundation to get easy access to advanced monographs. This textbook, tailored to the needs of graduate and advanced undergraduate students, covers Brownian motion, starting from its elementary properties, certain distributional aspects, path properties, and leading to stochastic calculus based on Brownian motion. It also includes numerical recipes for the simulation of Brownian motion.

## Probabilities on the Heisenberg Group

*Limit Theorems and Brownian Motion*

**Author**: Daniel Neuenschwander

**Publisher:**Springer

**ISBN:**3540685901

**Category:**Mathematics

**Page:**148

**View:**6845

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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.