Probability with Martingales

Author: David Williams
Publisher: Cambridge University Press
ISBN: 9780521406055
Format: PDF, ePub
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This is a masterly introduction to the modern and rigorous theory of probability. The author adopts the martingale theory as his main theme and moves at a lively pace through the subject's rigorous foundations. Measure theory is introduced and then immediately exploited by being applied to real probability theory. Classical results, such as Kolmogorov's Strong Law of Large Numbers and Three-Series Theorem are proved by martingale techniques. A proof of the Central Limit Theorem is also given. The author's style is entertaining and inimitable with pedagogy to the fore. Exercises play a vital role; there is a full quota of interesting and challenging problems, some with hints.

Brownian Motion

Author: René L. Schilling
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110307308
Format: PDF, ePub
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Stochastic processes occur everywhere in sciences and engineering, and need to be understood by applied mathematicians, engineers and scientists alike. This is a first course introducing the reader gently to the subject. Brownian motions are a stochastic process, central to many applications and easy to treat.

Probability and Stochastic Processes

Author: Ionut Florescu
Publisher: John Wiley & Sons
ISBN: 1118593138
Format: PDF, ePub
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A comprehensive and accessible presentation of probabilityand stochastic processes with emphasis on key theoretical conceptsand real-world applications With a sophisticated approach, Probability and StochasticProcesses successfully balances theory and applications in apedagogical and accessible format. The book’s primary focusis on key theoretical notions in probability to provide afoundation for understanding concepts and examples related tostochastic processes. Organized into two main sections, the book begins by developingprobability theory with topical coverage on probability measure;random variables; integration theory; product spaces, conditionaldistribution, and conditional expectations; and limit theorems. Thesecond part explores stochastic processes and related conceptsincluding the Poisson process, renewal processes, Markov chains,semi-Markov processes, martingales, and Brownian motion. Featuringa logical combination of traditional and complex theories as wellas practices, Probability and Stochastic Processes alsoincludes: Multiple examples from disciplines such as business,mathematical finance, and engineering Chapter-by-chapter exercises and examples to allow readers totest their comprehension of the presented material A rigorous treatment of all probability and stochasticprocesses concepts An appropriate textbook for probability and stochastic processescourses at the upper-undergraduate and graduate level inmathematics, business, and electrical engineering, Probabilityand Stochastic Processes is also an ideal reference forresearchers and practitioners in the fields of mathematics,engineering, and finance.

A Natural Introduction to Probability Theory

Author: R. Meester
Publisher: Springer Science & Business Media
ISBN: 9783764387242
Format: PDF, ePub
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Compactly written, but nevertheless very readable, appealing to intuition, this introduction to probability theory is an excellent textbook for a one-semester course for undergraduates in any direction that uses probabilistic ideas. Technical machinery is only introduced when necessary. The route is rigorous but does not use measure theory. The text 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. Only first-year calculus is required.

A First Look at Rigorous Probability Theory

Author: Jeffrey Seth Rosenthal
Publisher: World Scientific
ISBN: 9812703705
Format: PDF
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Features an introduction to probability theory using measure theory. This work provides proofs of the essential introductory results and presents the measure theory and mathematical details in terms of intuitive probabilistic concepts, rather than as separate, imposing subjects.

Probability Theory

Author: Yuan Shih Chow
Publisher: Springer Science & Business Media
ISBN: 1461219507
Format: PDF, ePub, Mobi
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Comprising the major theorems of probability theory and the measure theoretical foundations of the subject, the main topics treated here are independence, interchangeability, and martingales. Particular emphasis is placed upon stopping times, both as tools in proving theorems and as objects of interest themselves. No prior knowledge of measure theory is assumed and a unique feature of the book is the combined presentation of measure and probability. It is easily adapted for graduate students familiar with measure theory using the guidelines given. Special features include: - A comprehensive treatment of the law of the iterated logarithm - The Marcinklewicz-Zygmund inequality, its extension to martingales and applications thereof - Development and applications of the second moment analogue of Walds equation - Limit theorems for martingale arrays; the central limit theorem for the interchangeable and martingale cases; moment convergence in the central limit theorem - Complete discussion, including central limit theorem, of the random casting of r balls into n cells - Recent martingale inequalities - Cram r-L vy theorem and factor-closed families of distributions.

Martingale und Prozesse

Author: René L. Schilling
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110350688
Format: PDF
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Dieser Band ist der dritte Teil der „Modernen Stochastik". Als Fortsetzung der „Wahrscheinlichkeit" werden nun dynamische stochastische Phänomene anhand stochastischer Prozesse in diskreter Zeit betrachtet. Die erste Hälfte des Buchs gibt eine Einführung in die Theorie der diskreten Martingale – ihr Konvergenzverhalten, optional sampling & stopping, gleichgradige Integrierbarkeit und Martingalungleichungen. Die Stärke der Martingaltechniken wird in den Kapiteln über Anwendungen in der klassischen Wahrscheinlichkeitsrechnung und über die Burkholder-Davis-Gundy-Ungleichungen illustriert. Die zweite Hälfte des Buchs beschäftigt sich mit Irrfahrten auf dem Gitter Zd und auf Rd, ihrem Fluktuationsverhalten, Rekurrenz und Transienz. Die letzten beiden Kapitel geben einen Einblick in die probabilistische Potentialtheorie sowie einen Ausblick auf die Brownsche Bewegung: Donskers Invarianzprinzip. Contents Fair Play Bedingte Erwartung Martingale Stoppen und Lokalisieren Konvergenz von Martingalen L2-Martingale Gleichgradig integrierbare Martingale Einige klassische Resultate der W-Theorie Elementare Ungleichungen für Martingale Die Burkholder–Davis–Gundy Ungleichungen Zufällige Irrfahrten auf Zd – erste Schritte Fluktuationen einer einfachen Irrfahrt auf Z Rekurrenz und Transienz allgemeiner Irrfahrten Irrfahrten und Analysis Donskers Invarianzprinzip und die Brownsche Bewegung

Advanced Mathematical Tools for Automatic Control Engineers Volume 2

Author: Alex Poznyak
Publisher: Elsevier
ISBN: 9780080914039
Format: PDF, ePub
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Advanced Mathematical Tools for Automatic Control Engineers, Volume 2: Stochastic Techniques provides comprehensive discussions on statistical tools for control engineers. The book is divided into four main parts. Part I discusses the fundamentals of probability theory, covering probability spaces, random variables, mathematical expectation, inequalities, and characteristic functions. Part II addresses discrete time processes, including the concepts of random sequences, martingales, and limit theorems. Part III covers continuous time stochastic processes, namely Markov processes, stochastic integrals, and stochastic differential equations. Part IV presents applications of stochastic techniques for dynamic models and filtering, prediction, and smoothing problems. It also discusses the stochastic approximation method and the robust stochastic maximum principle. Provides comprehensive theory of matrices, real, complex and functional analysis Provides practical examples of modern optimization methods that can be effectively used in variety of real-world applications Contains worked proofs of all theorems and propositions presented

Wahrscheinlichkeitstheorie und Stochastische Prozesse

Author: Michael Mürmann
Publisher: Springer-Verlag
ISBN: 364238160X
Format: PDF, ePub, Mobi
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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 for Statistics and Machine Learning

Author: Anirban DasGupta
Publisher: Springer Science & Business Media
ISBN: 9781441996343
Format: PDF, Mobi
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This book provides a versatile and lucid treatment of classic as well as modern probability theory, while integrating them with core topics in statistical theory and also some key tools in machine learning. It is written in an extremely accessible style, with elaborate motivating discussions and numerous worked out examples and exercises. The book has 20 chapters on a wide range of topics, 423 worked out examples, and 808 exercises. It is unique in its unification of probability and statistics, its coverage and its superb exercise sets, detailed bibliography, and in its substantive treatment of many topics of current importance. This book can be used as a text for a year long graduate course in statistics, computer science, or mathematics, for self-study, and as an invaluable research reference on probabiliity and its applications. Particularly worth mentioning are the treatments of distribution theory, asymptotics, simulation and Markov Chain Monte Carlo, Markov chains and martingales, Gaussian processes, VC theory, probability metrics, large deviations, bootstrap, the EM algorithm, confidence intervals, maximum likelihood and Bayes estimates, exponential families, kernels, and Hilbert spaces, and a self contained complete review of univariate probability.