Book Description
This new book provides a unified, in-depth, readable introduction to the multipredictor regression methods most widely used in biostatistics: linear models for continuous outcomes, logistic models for binary outcomes, the Cox model for right-censored survival times, repeated-measures models for longitudinal and hierarchical outcomes, and generalized linear models for counts and other outcomes.
Treating these topics together takes advantage of all they have in common. The authors point out the many-shared elements in the methods they present for selecting, estimating, checking, and interpreting each of these models. They also show that these regression methods deal with confounding, mediation, and interaction of causal effects in essentially the same way.
The examples, analyzed using Stata, are drawn from the biomedical context but generalize to other areas of application. While a first course in statistics is assumed, a chapter reviewing basic statistical methods is included. Some advanced topics are covered but the presentation remains intuitive. A brief introduction to regression analysis of complex surveys and notes for further reading are provided. For many students and researchers learning to use these methods, this one book may be all they need to conduct and interpret multipredictor regression analyses.
The authors are on the faculty in the Division of Biostatistics, Department of Epidemiology and Biostatistics, University of California, San Francisco, and are authors or co-authors of more than 200 methodological as well as applied papers in the biological and biomedical sciences. The senior author, Charles E. McCulloch, is head of the Division and author of Generalized Linear Mixed Models (2003), Generalized, Linear, and Mixed Models (2000), and Variance Components (1992).
From the reviews:
"This book provides a unified introduction to the regression methods listed in the title...The methods are well illustrated by data drawn from medical studies...A real strength of this book is the careful discussion of issues common to all of the multipredictor methods covered."
Journal of Biopharmaceutical Statistics, 2005
"This book is not just for biostatisticians. It is, in fact, a very good, and relatively nonmathematical, overview of multipredictor regression models. Although the examples are biologically oriented, they are generally easy to understand and follow...I heartily recommend the book"
Technometrics, February 2006
"Overall, the text provides an overview of regression methods that is particularly strong in its breadth of coverage and emphasis on insight in place of mathematical detail. As intended, this well-unified approach should appeal to students who learn conceptually and verbally."
Journal of the American Statistical Association, March 2006
Customer Reviews:
very good book, compact but comprehensive.......2007-05-12
This book covers a wide range of topics in Biostatistics, in a comprehensive, but not overwhelming way. In my opinion this book has the potential of being useful to a broad audience, from Statisticians to other professionals who do health related research.
Excellent book ..........2007-01-09
A very specific book, with a lot of details for a statistitian
Book Description
PROBABILITY AND MEASURE
Third Edition
Now in its new third edition, Probability and Measure offers advanced students, scientists, and engineers an integrated introduction to measure theory and probability. Retaining the unique approach of the previous editions, this text interweaves material on probability and measure, so that probability problems generate an interest in measure theory and measure theory is then developed and applied to probability. Probability and Measure provides thorough coverage of probability, measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability, and stochastic processes. The Third Edition features an improved treatment of Brownian motion and the replacement of queuing theory with ergodic theory.
Like the previous editions, this new edition will be well received by students of mathematics, statistics, economics, and a wide variety of disciplines that require a solid understanding of probability theory.
Customer Reviews:
An exceptionally good book.......2006-12-19
I've read portions of almost every measure theoretic probability theory book published. And I've come back to Billingsley. This is a hard book to read through and through if you are a novice; this is not Billingsley's fault - it is just that the subject is hard on first acquaintance.
Billinglsey develops everything from first principles, so if you have the intellectual gumption you ought to be able to read the main text with a knowledge of plain college algebra and a little epsilon-delta practice of the sort that comes from an undergraduate real analysis course. The small print asides are fascinating but they are often addressed to a card carrying mathematician. The November 2003 reviewer who complained that Billingsley uses expectation before defining the integral fails to notice - or at any rate, to point out - that he defines only the expectation of simple random variables in the first chapter, so what is involved is just a sum, not an integral. I could sing my praises on and on. But here is the kernel of this review in a line: this is one of the best books ever written on measure theoretic probability. Full stop.
The book on probability.......2006-01-26
This book is not for everybody. It is for the professional mathematician (or physicist, or alike). All concepts are very well explained, and Billigsley does go down to the core of everything. It is, as far as I'm concerned, among the best books in math ever written, with favorites such as Feynman's lectures and Herstein's algebra manual. If you are a mathematician and want to have the top reference in probability, this is it.
Some nice examples, poorly organized.......2003-11-27
This is probably a very nice text book if you already know probability. There are undeniably some insightful examples. However, it is often hard to follow the sequence of topics in the book.
It is at least amusing that the integral is only developed a couple of chapters after expectation has been in use...
a very good text book.......2003-09-12
This book gives abundant examples and statements that help to deepen your understanding. It does not require much statistic background to follow the book, although sometimes the reasoning is not that obvious to me. But maybe because I am not a math student. I also feel that topics are a bit scattered in the book.
Standard text but ..........2003-06-19
the main problem with Billingsley's book lies in its organization of topics and results.
Yes it has all the standard results that need to be covered in a first (rigorous) course on probability theory and the proofs and exercises are good (thats why the three stars) but it is incredibly hard to study them from this book because of poor organisation which makes for lack of continuity (thats why no more than three).
Stick to Chung (and move to something more specialized thereafter). Unfortunately, Parthasarathy's 1977 Macmillan book is now out of print and only available in libraries ... I find that to be the best book at this level.
Book Description
This introduction to Probability Theory can be used, at the beginning graduate level, for a one-semester course on Probability Theory or for self-direction without benefit of a formal course; the measure theory needed is developed in the text. It will also be useful for students and teachers in related areas such as Finance Theory (Economics), Electrical Engineering, and Operations Research. The text covers the essentials in a directed and lean way with 28 short chapters. Assuming of readers only an undergraduate background in mathematics, it brings them from a starting knowledge of the subject to a knowledge of the basics of Martingale Theory. After learning Probability Theory from this text, the interested student will be ready to continue with the study of more advanced topics, such as Brownian Motion and Ito Calculus, or Statistical Inference. The second edition contains some additions to the text and to the references and some parts are completely rewritten.
Customer Reviews:
All background needed for Ito calculus is here.......2000-10-04
This is an excellent and timely textbook on probability and martingale theory. There is an increasing need of thorough but concise treatise of probability theory for researchers and graduate students in Engineering, Economics, Statistics and Mathematical Biology. Very few textbook fill this need. Jacod and Protter succeeded in bringing together essential concepts and theorems in probability/martingale theory in a clear and lucid style and the book is completely self-contained: all necessary machinery from measure theory are explained and proved while providing a flavor of probabilistic way of thinking. Unlike Williams' "Probability with Martingales", all mathematical details are covered in the body of text. They present conditional expectation through Hilbert space approach and Radon-Nikodym theorem is proved at the end of the book using martingales. This is an indoctrinated way of showing how martingales are applied in other field of mathematics. Each chapter starts with pedagogical explanation of concept and summary of results. This helps reader grasp concepts and develop intuition. The topics, examples and exercises are carefully chosen and well organized. I found several but minor typos and discrepancy in the notation during the last five chapters. Yes, elegant proof is given for each theorem on martingales but rephrasing them may help make it clear where in the proof previous results are used and applied. Also, it would be a great idea to include introductory texts on stochastic calculus in the reference for the beginning students. Despite these minor suggestions, I recommend the book with enthusiasm. After reading this book, one can take their way immediately to stochastic calculus: Brownian motion and Ito calculus and their applications.
All backgound needed for Ito calculus is here!.......2000-10-04
This is an excellent and timely textbook on probability and martingale theory. There is an increasing need of thorough but concise treatise of probability theory for researchers and graduate students in Engineering, Economics, Statistics and Mathematical Biology. Very few textbook fill this need. Jacod and Protter succeeded in bringing together essential concepts and theorems in probability/martingale theory in a clear and lucid style and the book is completely self-contained: all necessary machinery from measure theory are explained and proved while providing a flavor of probabilistic way of thinking. Unlike Williams' "Probability with Martingales", all mathematical details are covered in the body of text. They present conditional expectation through Hilbert space approach and Radon-Nikodym theorem is proved at the end of the book using martingales. This is an indoctrinated way of showing how martingales are applied in other field of mathematics. Each chapter starts with pedagogical explanation of concept and summary of results. This helps reader grasp concepts and develop intuition. The topics, examples and exercises are carefully chosen and well organized. I found several but minor typos and discrepancy in the notation during the last five chapters. Yes, elegant proof is given for each theorem on martingales but rephrasing them may help make it clear where in the proof previously results are used and applied. Also, it would be a great idea to include introductory texts on stochastic calculus for the beginning students. Despite these minor suggestions, I recommend the book with enthusiasm. After reading this book, one can take their way immediately to stochastic calculus: Brownian motion and Ito calculus.
Book Description
The Wiley Classics Library consists of selected books that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hopes to extend the life of these important works by making them available to future generations of mathematicians and scientists. Currently available in the Series: T. W. Anderson The Statistical Analysis of Time Series T. S. Arthanari & Yadolah Dodge Mathematical Programming in Statistics Emil Artin Geometric Algebra Norman T. J. Bailey The Elements of Stochastic Processes with Applications to the Natural Sciences Robert G. Bartle The Elements of Integration and Lebesgue Measure George E. P. Box & George C. Tiao Bayesian Inference in Statistical Analysis R. W. Carter Simple Groups of Lie Type William G. Cochran & Gertrude M. Cox Experimental Designs, Second Edition Richard Courant Differential and Integral Calculus, Volume I Richard Courant Differential and Integral Calculus, Volume II Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume I Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume II D. R. Cox Planning of Experiments Harold M. S. Coxeter Introduction to Modern Geometry, Second Edition Charles W. Curtis & Irving Reiner Representation Theory of Finite Groups and Associative Algebras Charles W. Curtis & Irving Reiner Methods of Representation Theory with Applications to Finite Groups and Orders, Volume I Charles W. Curtis & Irving Reiner Methods of Representation Theory with Applications to Finite Groups and Orders, Volume II Bruno de Finetti Theory of Probability, Volume 1 Bruno de Finetti Theory of Probability, Volume 2 W. Edwards Deming Sample Design in Business Research Amos de Shalit & Herman Feshbach Theoretical Nuclear Physics, Volume 1 —Nuclear Structure J. L. Doob Stochastic Processes Nelson Dunford & Jacob T. Schwartz Linear Operators, Part One, General Theory Nelson Dunford & Jacob T. Schwartz Linear Operators, Part Two, Spectral Theory—Self Adjoint Operators in Hilbert Space Nelson Dunford & Jacob T. Schwartz Linear Operators, Part Three, Spectral Operators Herman Feshbach Theoretical Nuclear Physics: Nuclear Reactions Bernard Friedman Lectures on Applications-Oriented Mathematics Phillip Griffiths & Joseph Harris Principles of Algebraic Geometry Gerald J. Hahn & Samuel S. Shapiro Statistical Models in Engineering Morris H. Hansen, William N. Hurwitz & William G. Madow Sample Survey Methods and Theory, Volume I—Methods and Applications Morris H. Hansen, William N. Hurwitz & William G. Madow Sample Survey Methods and Theory, Volume II—Theory Peter Henrici Applied and Computational Complex Analysis, Volume 1—Power Series—Integration—Conformal Mapping—Location of Zeros Peter Henrici Applied and Computational Complex Analysis, Volume 2—Special Functions—Integral Transforms—Asymptotics—Continued Fractions Peter Henrici Applied and Computational Complex Analysis, Volume 3—Discrete Fourier Analysis—Cauchy Integrals—Construction of Conformal Maps—Univalent Functions Peter Hilton & Yel-Chiang Wu A Course in Modern Algebra Harry Hochstadt Integral Equations Erwin O. Kreyszig Introductory Functional Analysis with Applications William H. Louisell Quantum Statistical Properties of Radiation Ali Hasan Nayfeh Introduction to Perturbation Techniques Emanuel Parzen Modern Probability Theory and Its Applications P. M. Prenter Splines and Variational Methods Walter Rudin Fourier Analysis on Groups C. L. Siegel Topics in Complex Function Theory, Volume I—Elliptic Functions and Uniformization Theory C. L. Siegel Topics in Complex Function Theory, Volume II—Automorphic and Abelian Integrals C. L. Siegel Topics in Complex Function Theory, Volume III—Abelian Functions & Modular Functions of Several Variables J. J. Stoker Differential Geometry J. J. Stoker Water Waves: The Mathematical Theory with Applications J. J. Stoker Nonlinear Vibrations in Mechanical and Electrical Systems
Customer Reviews:
A good introduction: concise and clear........2007-01-27
The book is concise and easy to follow. The author rarely gives lengthy explanations and analogies, but spends the bulk of the book stating solid facts and proofs. I also like the organization of the book. All definitions and theorems are explicitly stated and indexed, not scattered in paragraphs in the body of the text.
The book misses subjects such as complex measures (they are briefly mentioned), the fundamental theorem of calculus under Lebesgue settings, and probability measures, but its ok since the book is an introduction to the subject. A more comprehensive (and harder to read) book is "Real & Complex Analysis" by Walter Rudin. If you are interested in probability, consider Ptrick Billingsley's book "Probability and Measure".
Good Integration and Measure Into (A Bit Expensive Though).......2005-01-15
The exposition of integration in this book is the clearest I have read. I also found the chapter on modes of convergence, where it laid out the relationship between things such as L^P-convergence and convergence in measure, to be extremely useful. The second half, where it covers topics like Lebesgue measure, repeats some of the same information from the first part which is a bit iritating if you are reading straight throught, but contains a lot of good information. The book is also quite small making it easy to take with you as a quick reference.
Let me warn you though that this is an introduction to integration and measure _not_ an introduction to real analysis. It does not cover important topics like L^P-approximation, differentiation, etc. For a complete treatment of real analysis, I recommend the books "Lebesgue Integration on Euclidean Space" by Frank Jones and the slightly more abstract "Real and Functional Analysis" by Serge Lange.
IF YOU WANT TO UNDERSTAND MEASURE THEORY..........2001-06-04
IF YOU WANT TO UNDERSTAND MEASURE THEORY READ THIS BOOK, MAYBE THE ONLY PROBLEM IS THE LACK OF EXAMPLES BUT THE WAY THAT THE THEORY IS PRESENTED MAKE IT YOUR FIRST CHOICE WHEN YOU TRY TO LEARN MEASURE THEORY.
Excellent as an itroduction and as a reference.......2000-03-31
When I took my first one-semester course on measure and Lebesgue integration my teacher chose Bartle's "The Elements of Integration" as text. After reading many other books on the subject now I'm sure he made a wise decision.
Assuming almost no strong mathematical background, Bartle is able to build up the basic Lebesgue integral theory introducing the fundamental abstract concepts (sigma-algebra, measurable function, measure space, "almost everywhere", step function, etc.) in such an easy way that the student is not only able to handle them but to UNDERSTAND them.
From the first part of the book I appreciate specially chapters 6, 7, and 10, on L_p spaces, modes of convergence, and product measures, respectively. These chapters contain the most used results of the basic theory, and they are stated exactly in the way one needs them, making the book very useful for future reference.
I like the second part very much also, because it stresses the importance of measure theory by itself and not only as a requisite for integration theory. If you are interested in fractal geometry or geometric measure theory you will find chapters 11 to 17 very helpful.
Since I own this book it has never been lazy in my bookshelf.
A great place to begin.......2000-02-04
Measure and Integration is a daunting subject for mathematical neophytes. Bartle's little volume is the right place to start. I first learned measure theory from it 20 years ago and went on to study functional analysis and stochastic approximation.
I was able to master the material on my own with this book. The problems are at the right level and he begins with the correct level of abstraction. I recommend it over anything else because it is straighforward, clear and focused. Master it then go on to Walter Rudin's Real and Complex Analysis.
Book Description
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.
Customer Reviews:
interesting perspective.......2007-09-22
I have bought a lot of books from Amazon, but this is the first time I write a review. The reason is pretty obvious. I spent long hours learning probability theory from carefully reading Billingsley, which is worth every bit of effort I put in. Then, by chance I picked up this book from the library, and could not put it down.
Contrary to an earlier reviewer, I appreciate very much the same symbol for probability and expectation, which allows a quite natural and united treatment of the two important concepts. It brings great clarity.
The tempo of the book is at once deliberate and brisk. The author makes excellent judgment selecting the coverage. He also makes good decision on when to slow down and get dirty, and when to be brief and cover territory. The discussions in the book, combining simple explanation motivation and intuition, provides exciting road map for exploring in the first reading and for surveying in later readings.
Excellent book!
Excellent and idiosyncratic.......2005-05-11
There is no shortage of graduate-level probability textbooks. The classics (Ash, Billingsley, Breiman) have been partly replaced by (among others) Dudley, Shiryaev, and Durrett (the de facto gold standard). You can now Pollard's book to the list. It is written in a peculiar style, conversational yet rigorous. The author does not hide his preferences and feelings toward theorems, and I find this useful and illuminating, as it helps the reader sort the essential material from the ancillary. Most importantly, the choice of topics is truly unique. Clearly, the goal is to cover the basics very well, rather than offer an assortment of theorems. For example, the ergodic theorem (a mainstay of every textbook) is nowhere to be found. However, you can find advanced material that are becoming important tools for the applied probabilist (esp. the of the mathematical statistics variety) as well as advanced applications. Random examples: isoperimetric inequality, Dobb's theorem on consistency of posterior measures, coupling, multivariate normal distributions. Overall, I have read half of the book, and loved it. In my list of personal favorites, it ranks second behind William's booklet ("Probability with Martingales"). Like William's book, this is a quick, enjoyable, rigorous introduction to probabilistic tools. I am perfectly comfortable with a selection of topics, as long they are covered rigorously, they are motivated, and their relative importance is stressed. When this happens, the reader is well equipped to read a monography (e.g., Karatzas & Shreve) or a reference book (e.g., Kallenberg) for the in-depth study that subjects like ergodic theorem or brownian motion require.
Not recommended for beginners.......2005-03-03
This is an interesting book. It presents a rather complicated and difficult subject in a chatty un-math-book like way. I personally like that some of the time. I feel, however, that some things are just explained better when the presentation is more straightforward. The author jumps around a lot. I would really not recommend it for someone new to the subject. Get a more standard text. This book is better for someone who is already aquainted with the subject somewhat. Once you already know about the topics you can come back to this book and learn a lot more from it. But then again that is almost always true. What is the best book on probability? The fourth one you read. ;-D
fantastic but very difficult.......2005-02-18
i took probability theory with prof pollard using this book. at the time, i found the book to be very difficult and frustrating. i don't know if i would recommand it to the casual reader. however if you're really interested in probability theory and analysis in general, this is definitely the companion for you. indeed, after many a readings, i've found it to become an indispensable friend. it'll defnitely change the way you think about mathematics.
Worst book ever.......2004-12-09
This is the worst book you will find on measure theory and probability. The author does not provide any good explaination of any topic, and seem to be writing a comic book rather than a math book. Measure theory, which is covered in chapter-2 doesn't explain even the very basics and the actual content is put in an appendix. Furthermore, the author assumes that the reader is not interested in masure theory, so he resorts telling a lot of catch phrases. If you are writing a book, you should not hand wave and write jokes where mathematical arguments are needed.
Another annoying aspect of this book is that the author uses the P symbol for both probability and expectation. I found it very confusing (and contrary to author's claim that it "aids understanding," it only got me more frustrated.)
Probability and measure theory are simple concepts, if explained probability. For some reason, most books try to complicate things. I used to have great respect for Cambride University Press, but I don't know how this book got through their editorial process.
Book Description
Measure, Integral and Probability is a gentle introduction that makes measure and integration theory accessible to the average third-year undergraduate student. The ideas are developed at an easy pace in a form that is suitable for self-study, with an emphasis on clear explanations and concrete examples rather than abstract theory. For this second edition, the text has been thoroughly revised and expanded. New features include: · a substantial new chapter, featuring a constructive proof of the Radon-Nikodym theorem, an analysis of the structure of Lebesgue-Stieltjes measures, the Hahn-Jordan decomposition, and a brief introduction to martingales · key aspects of financial modelling, including the Black-Scholes formula, discussed briefly from a measure-theoretical perspective to help the reader understand the underlying mathematical framework. In addition, further exercises and examples are provided to encourage the reader to become directly involved with the material.
Customer Reviews:
absolutely useless.......2007-07-05
It starts out okay, good overview of measurable sets and the like. However, it does not even have the essential core theorem to the discipline stating when it is possible to integrate a function! one of the great thing about Lebesgue integration is that a function is integrebale in this sense IF AND ONLY IF the function is measurable. thats the whole point of having measurable functions. there is no if and only if theorem for RS integration. Plus other things, like it talks vaguely about 'randomly choosing a point' but with no precise definition. Things like that.
You are better off buying a classic by Halsey Royden or Walter Rudin, or something like that. This book is useless.
Excellent Book.......2007-06-27
The text is written at a level which is suitable for the classroom or self-teaching by an advanced student. The authors spare few details. I am very satisfied with my purchase.
Very good introduction to measure theory.......2007-04-13
Very good intro for first encounters with measure theory. Throughout the application in probability theory is emphasized. The necessity of each concept introduced is motivated with clear examples. Interesting problem sets are provided after each section; their solutions are given in the appendix.
good book.......2007-02-06
This a good book but will be a bit difficult for engineering graduates like me.. Should know real analysis and set theory to venture into this one.
clear introduction to measury theory.......2006-11-09
very clear measury theory introduction....with many detail solution to exercise....not bad !!
Average customer rating:
- Good
- Excellent and rigurous
- Good, but needs considerable background
- The best introduction to probability and measure
- Exceptionally Clear
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Probability And Measure Theory
Robert B. Ash
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ASIN: 0120652021 |
Book Description
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
Customer Reviews:
Good.......2007-09-29
The information in this book is so concise. The first two chapters are good for measure and integration theory.
Excellent and rigurous.......2007-02-28
This book deal with the whole picture of probability. One learns the very first roots of rigurous probability. And when I say rigurous I am not regarding it as "engineers rigour = nothing" but as "mathematicians rigour". The book is self-contained, the exposition is clear and is organized in the mathematic classical fashion: definition, lema, proof, theorem, proof.
That rigour, when it comes to probability beyond "number of successful cases / total number of cases", can only be achieved when the theory is developed in the most general background: measure theory. This gives general tools (theorems) which are applied to measures in general, a particular case of which is probability. Measure theory and general abstract Lebesgue integration go together, so the book defines and deepens in Lebesgue theory: integration, convergence theorems, Fubini's theorem, etc.
Once you feel confident and capable of deal with almost anything regarding Lebesgue integration the books moves on relations between integrals and measures: the Radon-Nikodym theorem which is perhaps one of the most important results of the book and whose proof is outstanding. It provides the reader with the tools to tackle Lebesgue almost everywhere differentiation theorem and absolutely continuous measures and functions.
Finally, before starting with probability as special case, there is a functional analysis chapter which gives proof of the three most important theorems of functional analisys in Hilbert and Banach spaces.
From chapter 4 on, everything about probability is covered. From basic distributions to martingales, ergodicity or central limit theoroem. But instead of making up ad-hoc theorems, theorems proved for measures in first chapters renders the proofs in this stage simply colloralies.
Once you read the book you will feel confident about anything touching probability, measure theory and Lebesgue integration and equipped with the most fundamental tools of functional analysis which are used widespread.
I couldn't recommend the book more.
Good, but needs considerable background.......2007-01-05
This was my textbook for a course in Probability Theory that I did in my third year at college. I had course work in Probability, but this course took a measure theoretic approach to probability. This book does the same. I found that the book is written for an audience that already understands some measure theory. That notwithstaning, I still think the book is an excellent introduction to Probability through measure, and is one of the most comprehensive books on the topic. Almost everything one might want to talk about in the subject are dealth with thoroughly. For first timers, the book is a little difficult to follow, but a little perseverance should pay off. This book is something every grad student of mathematics should have on his bookshelf. This also happens to be one of those rare math books that have a selection of the exercises solved at the end. Cant ask for more, can you?
I also recommend K L Chung's book on advanced probability. Sometimes when I was stuck with Ash, I referred to Chung.
The best introduction to probability and measure.......2005-02-09
The book very nicely develops the basics of measure theory from a probability perspective (e.g. includes Caratheodory extension theorem, Lebesgue-Stieltjes measures, weak convergence and Kolmogorov extension theorem). It then gives a brief introduction to functional analysis and proceeds to probability theory, martingales and concludes with brownian motion and stochastic integration.
All standard results are given and the book is self-contained. It is a concise, yet readable introduction to this area (less concise then Rudin, Williams but more than Billingsly). An excellent feature of this book is that full solutions to some of the exercises are provided at the end. This makes this book ideal for self-study. The only prerequisite for this book is elementary real analysis (say chapters 1-7 of Rudin's principles of mathematical analysis).
There are other excellent books on measure theory (Rudin, Royden), but if you are interested in measure theory from a probabilistic view this is the book to choose.
As far as a probability textbook, it is clearer and more readable than Billingsly, Chung, Williams and Durrett.
Exceptionally Clear.......2002-07-10
I first used this text in the earlier version, which comprises the first half of the book, in a one-year course in Hilbert Spaces and Lebesgue Measure theory when in the first year of grad school. The material is presented in a clearly written manner and the exposition is some of the clearest mathematical writing I've seen in a subject which is replete with textbooks.
Anyone who wants to be inaugurated into the "mysteries" of measure theory and the fine points of the rigorous theory of stochastic processes and the Ito integral, will do himself or herself a favor by using this text. If it is not assigned to your class and you have the extra cash, order it anyway. It is also well-suited for self-study.
Book Description
A new look at weak-convergence methods in metric spaces-from a master of probability theory In this new edition, Patrick Billingsley updates his classic work Convergence of Probability Measures to reflect developments of the past thirty years. Widely known for his straightforward approach and reader-friendly style, Dr. Billingsley presents a clear, precise, up-to-date account of probability limit theory in metric spaces. He incorporates many examples and applications that illustrate the power and utility of this theory in a range of disciplines-from analysis and number theory to statistics, engineering, economics, and population biology. With an emphasis on the simplicity of the mathematics and smooth transitions between topics, the Second Edition boasts major revisions of the sections on dependent random variables as well as new sections on relative measure, on lacunary trigonometric series, and on the Poisson-Dirichlet distribution as a description of the long cycles in permutations and the large divisors of integers. Assuming only standard measure-theoretic probability and metric-space topology, Convergence of Probability Measures provides statisticians and mathematicians with basic tools of probability theory as well as a springboard to the "industrial-strength" literature available today.
Customer Reviews:
THE BOOK on WEAK CONVERGENCE.......1998-07-16
A classic, will survive through the ages as long as Real Analysis and Probability are studied by students the world over. You need it to get a fundamental grounding in Probability Theory
Book Description
During the past two decades several remarkable new results were discovered about harmonic measure in the complex plane. This volume offers a careful survey of these results and an introduction to the branch of analysis that contains them. Many of these results, due to Bishop, Carleson, Jones, Makarov, Wolff, and others, appear here in book form for the first time. The first four chapters present the necessary background material on univalent functions, potential theory, and extremal length. In addition, each chapter has numerous exercises to further inform and instruct readers. This work is accessible to students who have completed standard graduate courses in real and complex analysis.
Book Description
This is a graduate level textbook on measure theory and probability theory. The book can be used as a text for a two semester sequence of courses in measure theory and probability theory, with an option to include supplemental material on stochastic processes and special topics. It is intended primarily for first year Ph.D. students in mathematics and statistics although mathematically advanced students from engineering and economics would also find the book useful. Prerequisites are kept to the minimal level of an understanding of basic real analysis concepts such as limits, continuity, differentiability, Riemann integration, and convergence of sequences and series. A review of this material is included in the appendix.
The book starts with an informal introduction that provides some heuristics into the abstract concepts of measure and integration theory, which are then rigorously developed. The first part of the book can be used for a standard real analysis course for both mathematics and statistics Ph.D. students as it provides full coverage of topics such as the construction of Lebesgue-Stieltjes measures on real line and Euclidean spaces, the basic convergence theorems, L^p spaces, signed measures, Radon-Nikodym theorem, Lebesgue's decomposition theorem and the fundamental theorem of Lebesgue integration on R, product spaces and product measures, and Fubini-Tonelli theorems. It also provides an elementary introduction to Banach and Hilbert spaces, convolutions, Fourier series and Fourier and Plancherel transforms. Thus part I would be particularly useful for students in a typical Statistics Ph.D. program if a separate course on real analysis is not a standard requirement.
Part II (chapters 6-13) provides full coverage of standard graduate level probability theory. It starts with Kolmogorov's probability model and Kolmogorov's existence theorem. It then treats thoroughly the laws of large numbers including renewal theory and ergodic theorems with applications and then weak convergence of probability distributions, characteristic functions, the Levy-Cramer continuity theorem and the central limit theorem as well as stable laws. It ends with conditional expectations and conditional probability, and an introduction to the theory of discrete time martingales.
Part III (chapters 14-18) provides a modest coverage of discrete time Markov chains with countable and general state spaces, MCMC, continuous time discrete space jump Markov processes, Brownian motion, mixing sequences, bootstrap methods, and branching processes. It could be used for a topics/seminar course or as an introduction to stochastic processes.
From the reviews:
"...There are interesting and non-standard topics that are not usually included in a first course in measture-theoretic probability including Markov Chains and MCMC, the bootstrap, limit theorems for martingales and mixing sequences, Brownian motion and Markov processes. The material is well-suported with many end-of-chapter problems." D.L. McLeish for Short Book Reviews of the ISI, December 2006
Customer Reviews:
Good Book But ..........2007-02-07
This is a good book. And the authors try to summary the measure theory and probability theory into one book. Benefit of doing this is easy to see relationship between the two theories more clearly than reading one for each topic. However, I should say that no book trying to do this job is successful, including this one. To my experience, better understand of real analysis is necessary. If you do not, I think this book is not suitable for you. If you do, you can start the book from chapter 6 and treat the 1-5 chapter as a good reference.
Moreover, the statements in this book are quite concise and I like this style. However, this is a quite new one. There are pretty much typos in the book. I expect that the second edition will be much better than this one.
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