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: 462

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

Modern Probability Theory

Modern Probability Theory

Gnedenko , B . V . ( 1976 ) , The Theory of Probability , Mir Publishers , Moscow .
Hall , P . , and Heyde , C . C . ( 1980 ) , Martingale Limit Theory and Its
Applications , Academic Press , N . Y . Halmos , P . R . ( 1958 ) , Measure Theory ,
Van ...

Author: B. Ramdas Bhat

Publisher: New Age International

ISBN: 8122411894

Category: Probabilities

Page: 344

View: 264

The Book Continues To Cover The Syllabus Of A One-Year Course On Probability Theory. The Rigorous Axiomatic Approach Continues To Be Followed. For Those Who Plan To Apply Probability Models In Their Chosen Areas The Book Will Provide The Necessary Foundation. For Those Who Want To Proceed To Work In The Area Of Stochastic Processes, The Present Work Will Provide The Necessary Preliminary Background. It Can Be Used By Probabilists, Statisticians And Mathematicians. In The Present Revised Edition Many Concepts Have Been Elaborated. Clarifications Are Given For A Number Of Steps In The Proofs Of Results Derived. Additional Examples And Problems Are Given At The End Of Different Chapters. An Additional Preliminary Chapter Has Been Added So That Students Can Recapitulate The Topics Normally Covered In The Undergraduate Courses. It Also Forms The Foundation For Topics Covered In The Remaining Chapters. The Third Edition Incorporates The Suggestions For Improvements Received By The Author When The Earlier Editions Were In Circulation. With The Additional Features And Most Of The Errors Weeded Out, The Book Is Hoped To Become More Useful In The Hands Of Students And Teachers.
Categories: Probabilities

Stochastic Processes

Stochastic Processes

Martingale Dynamics. New York: Springer-Verlag. Hall, P (1980). Martingale
Limit Theory and its Application. San Diego: Academic Press. Liptser, RS and AN
Shiryaev (1989). Theory of Martingales (Translated by K. Dzjaparidze). Dordrecht
: ...

Author: Narahari U Prabhu

Publisher: World Scientific Publishing Company

ISBN: 9789813106956

Category: Mathematics

Page: 356

View: 335

Most introductory textbooks on stochastic processes which cover standard topics such as Poisson process, Brownian motion, renewal theory and random walks deal inadequately with their applications. Written in a simple and accessible manner, this book addresses that inadequacy and provides guidelines and tools to study the applications. The coverage includes research developments in Markov property, martingales, regenerative phenomena and Tauberian theorems, and covers measure theory at an elementary level.
Categories: Mathematics

Stochastic Limit Theory

Stochastic Limit Theory

... Martingale Limit Theory and its Application (1980) is an important landmark, as
are a series of papers by econometricians including among others Halbert White,
Ronald Gallant, Donald Andrews, and Herman Bierens. This work introduced ...

Author: James Davidson

Publisher: OUP Oxford

ISBN: 9780191525049

Category: Business & Economics

Page: 562

View: 210

This is a survey of the recent developments in the rapidly expanding field of asymptotic distribution theory, with a special emphasis on the problems of time dependence and heterogeneity. The book is designed to be useful on two levels. First as a textbook and reference work, giving definitions of the relevant mathematical concepts, statements, and proofs of the important results from the probability literature, and numerous examples; and second, as an account of recent work in the field of particular interest to econometricians, including a number of important new results. It is virtually self-contained, with all but the most basic technical prerequisites being explained in their context; mathematical topics include measure theory, integration, metric spaces, and topology, with applications to random variables, and an extended treatment of conditional probability. Other subjects treated include: stochastic processes, mixing processes, martingales, mixingales, and near-epoch dependence; the weak and strong laws of large numbers; weak convergence; and central limit theorems for nonstationary and dependent processes. The functional central limit theorem and its ramifications are covered in detail, including an account of the theoretical underpinnings (the weak convergence of measures on metric spaces), Brownian motion, the multivariate invariance principle, and convergence to stochastic integrals. This material is of special relevance to the theory of cointegration.
Categories: Business & Economics

A User s Guide to Measure Theoretic Probability

A User s Guide to Measure Theoretic Probability

Dellacherie, C. & Meyer, P. A. (1982), Probabilities and Potential B: Theory of
Martingales, North-Holland, Amsterdam. Doob ... Hall, P. & Heyde, C. C. (1980),
Martingale Limit Theory and Its Application, Academic Press, New York, NY.
Hewitt ...

Author: David Pollard

Publisher: Cambridge University Press

ISBN: 0521002893

Category: Mathematics

Page: 351

View: 103

This book grew from a one-semester course offered for many years to a mixed audience of graduate and undergraduate students who have not had the luxury of taking a course in measure theory. The core of the book covers the basic topics of independence, conditioning, martingales, convergence in distribution, and Fourier transforms. In addition there are numerous sections treating topics traditionally thought of as more advanced, such as coupling and the KMT strong approximation, option pricing via the equivalent martingale measure, and the isoperimetric inequality for Gaussian processes. The book is not just a presentation of mathematical theory, but is also a discussion of why that theory takes its current form. It will be a secure starting point for anyone who needs to invoke rigorous probabilistic arguments and understand what they mean.
Categories: Mathematics

Probability For Analysts

Probability For Analysts

Karl Stromberg. (5) For other nice proofs of Lebesgue's Differentiation Theorem
via martingales see [19] and [21]. ... [14] HALL, P. and HEYDE, C. C., Martingale
Limit Theory and its Applications, Academic Press, New York, 1980. [15] Issac, R.

Author: Karl Stromberg

Publisher: CRC Press

ISBN: 0412041715

Category: Mathematics

Page: 330

View: 939

This book will enable researchers and students of analysis to more easily understand research papers in which probabilistic methods are used to prove theorems of analysis, many of which have no other known proofs. The book assumes a course in measure and integration theory but requires little or no background in probability theory. It emplhasizes topics of interest to analysts, including random series, martingales and Brownian motion.
Categories: Mathematics

Brownian Motion and Stochastic Calculus

Brownian Motion and Stochastic Calculus

Ioannis Karatzas, Steven Shreve. FRIEDMAN, A. (1976) Stochastic Differential
Equations and Applications, Volume 2. Academic Press ... HALL, P. & HEYDE,
C.C. (1980) Martingale Limit Theory and Its Application. Academic Press, New
York.

Author: Ioannis Karatzas

Publisher: Springer

ISBN: 9781461209492

Category: Mathematics

Page: 470

View: 920

A graduate-course text, written for readers familiar with measure-theoretic probability and discrete-time processes, wishing to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed, illustrated by results concerning representations of martingales and change of measure on Wiener space, which in turn permit a presentation of recent advances in financial economics. The book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The whole is backed by a large number of problems and exercises.
Categories: Mathematics

Large Sample Techniques for Statistics

Large Sample Techniques for Statistics

Hall and Heyde (1980) Martingale Limit Theory and Its Application 8.1
Introduction The term martingale originally referred to a betting strategy. Imagine
a gambler playing a blackjack game (also known as twenty—one) in a casino (if
you have ...

Author: Jiming Jiang

Publisher: Springer Science & Business Media

ISBN: 9781441968272

Category: Mathematics

Page: 610

View: 820

In a way, the world is made up of approximations, and surely there is no exception in the world of statistics. In fact, approximations, especially large sample approximations, are very important parts of both theoretical and - plied statistics.TheGaussiandistribution,alsoknownasthe normaldistri- tion,is merelyonesuchexample,dueto thewell-knowncentrallimittheorem. Large-sample techniques provide solutions to many practical problems; they simplify our solutions to di?cult, sometimes intractable problems; they j- tify our solutions; and they guide us to directions of improvements. On the other hand, just because large-sample approximations are used everywhere, and every day, it does not guarantee that they are used properly, and, when the techniques are misused, there may be serious consequences. 2 Example 1 (Asymptotic? distribution). Likelihood ratio test (LRT) is one of the fundamental techniques in statistics. It is well known that, in the 2 “standard” situation, the asymptotic null distribution of the LRT is?,with the degreesoffreedomequaltothe di?erencebetweenthedimensions,de?ned as the numbers of free parameters, of the two nested models being compared (e.g., Rice 1995, pp. 310). This might lead to a wrong impression that the 2 asymptotic (null) distribution of the LRT is always? . A similar mistake 2 might take place when dealing with Pearson’s? -test—the asymptotic distri- 2 2 bution of Pearson’s? -test is not always? (e.g., Moore 1978).
Categories: Mathematics

Theory of Martingales

Theory of Martingales

'; 'One service category theory has rendered mathematics ... '. All arguably true_ And all statements obtainable this way form part of the raison d'etre of this series_ This series, Mathematics and Its ApplicatiOns, started in 1977.

Author: Robert Liptser

Publisher: Springer Science & Business Media

ISBN: 9789400924383

Category: Mathematics

Page: 792

View: 720

One service mathematics has rc:ndered the 'Et moi, "', si j'avait su comment CD revenir, je n'y serais point alle. ' human race. It has put common SCIIJC back Jules Verne where it belongs. on the topmost shelf next to tbe dusty canister 1abdled 'discarded non- The series is divergent; tberefore we may be sense'. able to do sometbing witb it Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ... '; 'One service logic has rendered com puter science ... '; 'One service category theory has rendered mathematics ... '. All arguably true_ And all statements obtainable this way form part of the raison d'etre of this series_ This series, Mathematics and Its ApplicatiOns, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope_ At the time I wrote "Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches.
Categories: Mathematics

Statistical Methods for Stochastic Differential Equations

Statistical Methods for Stochastic Differential Equations

On continuous martingales. Proc. Nat. Acad. Sci. U.S.A., 53, 913–916. Duffie, D. (
1996). Dynamic Asset Pricing Theory. Princeton, N.J.: Princeton University ...
Martingale Limit Theory and Its Application. Boston: Academic Press. Harrison, M
.

Author: Mathieu Kessler

Publisher: CRC Press

ISBN: 9781439849408

Category: Mathematics

Page: 507

View: 286

The seventh volume in the SemStat series, Statistical Methods for Stochastic Differential Equations presents current research trends and recent developments in statistical methods for stochastic differential equations. Written to be accessible to both new students and seasoned researchers, each self-contained chapter starts with introductions to the topic at hand and builds gradually towards discussing recent research. The book covers Wiener-driven equations as well as stochastic differential equations with jumps, including continuous-time ARMA processes and COGARCH processes. It presents a spectrum of estimation methods, including nonparametric estimation as well as parametric estimation based on likelihood methods, estimating functions, and simulation techniques. Two chapters are devoted to high-frequency data. Multivariate models are also considered, including partially observed systems, asynchronous sampling, tests for simultaneous jumps, and multiscale diffusions. Statistical Methods for Stochastic Differential Equations is useful to the theoretical statistician and the probabilist who works in or intends to work in the field, as well as to the applied statistician or financial econometrician who needs the methods to analyze biological or financial time series.
Categories: Mathematics

Probability Theory

Probability Theory

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.

Author: Yuan Shih Chow

Publisher: Springer Science & Business Media

ISBN: 9781461219507

Category: Mathematics

Page: 489

View: 749

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.
Categories: Mathematics

Nonparametric Econometrics

Nonparametric Econometrics

Martingale limit theory and its application. Academic Press. Hall, P. and J. D. Hart
. (1990). Bootstrap test for difference between means in nonparametric
regression. Journal of the American Statistical Association 85, 1039–1049. Hall,
P. and ...

Author: Qi Li

Publisher: Princeton University Press

ISBN: 9781400841066

Category: Business & Economics

Page: 768

View: 935

Until now, students and researchers in nonparametric and semiparametric statistics and econometrics have had to turn to the latest journal articles to keep pace with these emerging methods of economic analysis. Nonparametric Econometrics fills a major gap by gathering together the most up-to-date theory and techniques and presenting them in a remarkably straightforward and accessible format. The empirical tests, data, and exercises included in this textbook help make it the ideal introduction for graduate students and an indispensable resource for researchers. Nonparametric and semiparametric methods have attracted a great deal of attention from statisticians in recent decades. While the majority of existing books on the subject operate from the presumption that the underlying data is strictly continuous in nature, more often than not social scientists deal with categorical data--nominal and ordinal--in applied settings. The conventional nonparametric approach to dealing with the presence of discrete variables is acknowledged to be unsatisfactory. This book is tailored to the needs of applied econometricians and social scientists. Qi Li and Jeffrey Racine emphasize nonparametric techniques suited to the rich array of data types--continuous, nominal, and ordinal--within one coherent framework. They also emphasize the properties of nonparametric estimators in the presence of potentially irrelevant variables. Nonparametric Econometrics covers all the material necessary to understand and apply nonparametric methods for real-world problems.
Categories: Business & Economics

Probability Theory

Probability Theory

Aimed primarily at graduate students and researchers, this text is a comprehensive course in modern probability theory and its measure-theoretical foundations.

Author: Achim Klenke

Publisher: Springer Science & Business Media

ISBN: 1848000480

Category: Mathematics

Page: 621

View: 227

Aimed primarily at graduate students and researchers, this text is a comprehensive course in modern probability theory and its measure-theoretical foundations. It covers a wide variety of topics, many of which are not usually found in introductory textbooks. The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in the world of probability theory. In addition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation.
Categories: Mathematics

Classical Potential Theory and Its Probabilistic Counterpart

Classical Potential Theory and Its Probabilistic Counterpart

An application of Theorem 4 shows that one version of GMx(-) is an almost surely
right continuous martingale, and in the ... is a downward-directed class of almost
surely right continuous martingales with essential order infimum and limit GMF, ...

Author: Joseph L. Doob

Publisher: Springer Science & Business Media

ISBN: 3540412069

Category: Mathematics

Page: 846

View: 931

From the reviews: "Here is a momumental work by Doob, one of the masters, in which Part 1 develops the potential theory associated with Laplace's equation and the heat equation, and Part 2 develops those parts (martingales and Brownian motion) of stochastic process theory which are closely related to Part 1". --G.E.H. Reuter in Short Book Reviews (1985)
Categories: Mathematics

Probability Theory an Analytic View

Probability Theory  an Analytic View

Revised edition of a first-year graduate course on probability theory.

Author: Daniel W. Stroock

Publisher: Cambridge University Press

ISBN: 0521663490

Category: Mathematics

Page: 536

View: 831

Revised edition of a first-year graduate course on probability theory.
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: 924

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

Weak Convergence and Its Applications

Weak Convergence and Its Applications

In this book, we will introduce some recent development of modern weak convergence theory to overcome defects of classical theory.Contents: "The Definition and Basic Properties of Weak Convergence: "Metric SpaceThe Definition of Weak ...

Author: Zhengyan Lin

Publisher: World Scientific

ISBN: 9789814447706

Category: Convergence

Page: 176

View: 891

Weak convergence of stochastic processes is one of most important theories in probability theory. Not only probability experts but also more and more statisticians are interested in it. In the study of statistics and econometrics, some problems cannot be solved by the classical method. In this book, we will introduce some recent development of modern weak convergence theory to overcome defects of classical theory.Contents: "The Definition and Basic Properties of Weak Convergence: "Metric SpaceThe Definition of Weak Convergence of Stochastic Processes and Portmanteau TheoremHow to Verify the Weak Convergence?Two Examples of Applications of Weak Convergence"Convergence to the Independent Increment Processes: "The Basic Conditions of Convergence to the Gaussian Independent Increment ProcessesDonsker Invariance PrincipleConvergence of Poisson Point ProcessesTwo Examples of Applications of Point Process Method"Convergence to Semimartingales: "The Conditions of Tightness for Semimartingale SequenceWeak Convergence to SemimartingaleWeak Convergence to Stochastic Integral I: The Martingale Convergence ApproachWeak Convergence to Stochastic Integral II: Kurtz and Protter's ApproachStable Central Limit Theorem for SemimartingalesAn Application to Stochastic Differential EquationsAppendix: The Predictable Characteristics of Semimartingales"Convergence of Empirical Processes: "Classical Weak Convergence of Empirical ProcessesWeak Convergence of Marked Empirical ProcessesWeak Convergence of Function Index Empirical ProcessesWeak Convergence of Empirical Processes Involving Time-Dependent dataTwo Examples of Applications in Statistics Readership: Graduate students and researchers in probability & statistics and econometrics.
Categories: Convergence

Probability with Martingales

Probability with Martingales

These measure-theoretic results are proved in full in appendices, so that the book is completely self-contained. The book is written for students, not for researchers, and has evolved through several years of class testing.

Author: David Williams

Publisher: Cambridge University Press

ISBN: 9781139642989

Category: Mathematics

Page:

View: 323

Probability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. This book is a modern, lively and rigorous account which has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's Strong Law of Large Numbers and the Three-Series Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions. A distinguishing feature is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals; and it assumes certain key results from measure theory in the main text. These measure-theoretic results are proved in full in appendices, so that the book is completely self-contained. The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital rôle. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction.
Categories: Mathematics

Probability

Probability

A well-written and lively introduction to measure theoretic probability for graduate students and researchers.

Author: Rick Durrett

Publisher: Cambridge University Press

ISBN: 9781108473682

Category: Mathematics

Page: 432

View: 579

A well-written and lively introduction to measure theoretic probability for graduate students and researchers.
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: 913

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