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.
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"Unlike technical books of a previous generation, here we have an author admitting that a reader might find the subject difficult and even offering a window on the pedagogical considerations by which he shapes his exposition. Pollard does not just explain and clarify abstractions; he really sells them to a presumably skeptical reader. Thus he bridges a gap in the literature, between elementary probability texts and advanced works that presume a secure prior knowledge of measure theory...The nice layout and occasional useful diagram further amplify the friendliness of this book." Choice
"The book ... can be recommended as an excellent source in measuring theoretic probability theory as well as a handbook for everybody who studies stochastic processes in the real world." Mathematical Reviews
"The book ... can be recommended as an excellent source in measuring theoretic probability theory as well as a handbook for everybody who studies stochastic processes in the real world." Mathematical Reviews
Book Description
Rigorous probabilistic arguments, built on the foundation of measure theory introduced seventy years ago by Kolmogorov, have invaded many fields. Many students of statistics, biostatistics, econometrics, finance, and other changing disciplines now find themselves needing to absorb theory beyond what they might have learned in the typical undergraduate, calculus-based probability course. This book grew from a one-semester course offered for many years to a mixed audience of graduate and undergraduate students, who were expected only to have taken an undergraduate course in real analysis or advanced calculus.
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Customer Reviews
Most helpful customer reviews
By ktrmes
Format:Paperback
I first saw this book online and decided that I would buy it when it became available. It is a very nice book -- one of those rare math books with material so well arranged that can be read almost like a novel. The material in Chapter 2 (A Modicum of Measure Theory) in particular is a fine example of this -- I had learned this material from Royden whose presentation, though providing just what a math graduate student should have, does not make the theorems part of a story in an historical context. This context, the level and pace of the presentation and the book's conscious acknowledgement of motivations make for a very readable presentation.
Format:Paperback
First off I must say we haven't had a publication in measure theory or abstract probability for decades which integrates as much specialty knowledge and wide range of application as Pollard's 2002 "A User's Guide to Measure Theoretic Probability" that is able to prove it! Previous to this work, all these unneccesary distinctions and misunderstandings have been made (and are still being made) between the discrete and the continuous in mathematics, and physics as well. Im not going to spoil the suprises on how it's done but will simply point out that this work should soon be prerequisite reading for all graduates moving on towards pure mathematics and general-unified field theoretic applications. Once we can get a concrete understanding of this work we may soon no longer teach nor practice probability theory and mathematics as separated theories nor as separated fields!
A User's Guide to Measure Theoretic Probability is a quality book, as are all the books in the Cambridge Series in Statistical and Probabilistic Mathematics (see Wavelet Methods for Time Series Analysis, the Determination and Tracking of Frequency, Bayesian Methods). Illustrations are included in the book as well. You can have a look at the book in PDF format on Pollard's website
http://www.stat.yale.edu/~pollard/
Topical contents of interest for this book include:
Reveals that independence of random variables by means of distribution functions can be done metricaly using product measures instead of factorizing joint densities and assuming independence as transformational smoothness. In other words you can actually do the math smoothly instead of generalizing it as such.
The discrete and the continuous no longer have to be taught at the graduate level as though they were differential. This fact is proven in Chapter III by means of derivation theory, decomposition (not loss but preservation/re-composition) of Lebesgue measure.
Univariate and multivariate no longer distinct? Once again the proofs are in the book. Distributions and joint distributions using grainy calculations, Jacobians, many integrals, and matrices, are easily achieved independantly (mind you with a bit of extra intuitive rigour) using measure theory. Distributions can be determined as image measures and joint distributions as image measures for mappings into product spaces discussed in Chapters II and IV.
The first two chapters introduce this beneficial application in probability with:
Measures and sigma-fields, Measureable functions, Integrals, Construction of integrals from measures, Limit theorems, Neglible sets, Lp-spaces, Uniform integrability, Image measures and distributions, Generating classes of sets and functions,
Chapter III is about Desnsities and Derivatives:
Densities and absolute continuity, The Lebesgue decomposition, Distances and affinities between measures, Classical concepts of absolute continuity, Vitali covering lemma, etc.
Chapter IV Product Spaces and Independence: Independence of sigma-fields, Construction of measures on product spaces, Product measures, Infinite product spaces etc.
there are 351 pages and chapters 1-4 only go to page 108. The rest of the book includes massive ammounts of data on: Conditional distributions, Integration and disintregration, Conditional densities, Invariance, Kolmogorov's abstact conditional expectation, Sufficiency. Chapter VI is all about Martingales; Stopping times, Convergence, Krickeberg decomposition, Uniform integrability, Reversed Martingales, Symmetry, Exchangeability, Lindeberg's method, Multivariate limit theorems, Fourier transforms, Martingale central limit theorem, the Levy and Cramer-Wold theorems, Brownian motion etc.
Also by Chapter 10 you get into: Representations and Couplings;
Strassen's theorem, the Yurinskii coupling, Quantile coupling of Binomials with normals, Haar coupling etc. Chapters 11 and 12: Exponential Tails and and the Law of the Iterated Logarithm; Identically distributed summands, Multivariate Normal Distributions; Ferniques inequality, and it's proof, Gaussian isoperimetric inequality, and it's prrof. The extensive Appendices include topics: Measures and inner measures, Tightness, Countable additivity, Extension to the ^c-closure, Integral representations, Hilbert Spaces; Orthogonal projections, Orthonormal bases, Series expansions, Convexity; Convex sets and functions, their Integral representations, their relative interiors, Quantile coupling of the Binomial with the normal, Martingales in Ccontinuous Time; using filtrations, Brownian filtrations, supermartingales etc. Disintegration of Measures; Representations of measures on product spaces, disintegrations with respect to a measurable map etc.
Most of the material covered has previously only been available in French contexts and we are lucky to have now in english, available to Phd's in the United States and written at the Graduate level for students worldwide.
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
4.1 out of 5 stars
7 reviews
28 of 31 people found the following review helpful
5.0 out of 5 stars
Finally a Revolutionary Ressolution for Mathematics!
Sep 26 2003
By jeremy.jae@cell.matrix.cn
- Published on Amazon.com
Format:Paperback
First off I must say we haven't had a publication in measure theory or abstract probability for decades which integrates as much specialty knowledge and wide range of application as Pollard's 2002 "A User's Guide to Measure Theoretic Probability" that is able to prove it! Previous to this work, all these unneccesary distinctions and misunderstandings have been made (and are still being made) between the discrete and the continuous in mathematics, and physics as well. Im not going to spoil the suprises on how it's done but will simply point out that this work should soon be prerequisite reading for all graduates moving on towards pure mathematics and general-unified field theoretic applications. Once we can get a concrete understanding of this work we may soon no longer teach nor practice probability theory and mathematics as separated theories nor as separated fields!
A User's Guide to Measure Theoretic Probability is a quality book, as are all the books in the Cambridge Series in Statistical and Probabilistic Mathematics (see Wavelet Methods for Time Series Analysis, the Determination and Tracking of Frequency, Bayesian Methods). Illustrations are included in the book as well. You can have a look at the book in PDF format on Pollard's website
http://www.stat.yale.edu/~pollard/
Topical contents of interest for this book include:
Reveals that independence of random variables by means of distribution functions can be done metricaly using product measures instead of factorizing joint densities and assuming independence as transformational smoothness. In other words you can actually do the math smoothly instead of generalizing it as such.
The discrete and the continuous no longer have to be taught at the graduate level as though they were differential. This fact is proven in Chapter III by means of derivation theory, decomposition (not loss but preservation/re-composition) of Lebesgue measure.
Univariate and multivariate no longer distinct? Once again the proofs are in the book. Distributions and joint distributions using grainy calculations, Jacobians, many integrals, and matrices, are easily achieved independantly (mind you with a bit of extra intuitive rigour) using measure theory. Distributions can be determined as image measures and joint distributions as image measures for mappings into product spaces discussed in Chapters II and IV.
The first two chapters introduce this beneficial application in probability with:
Measures and sigma-fields, Measureable functions, Integrals, Construction of integrals from measures, Limit theorems, Neglible sets, Lp-spaces, Uniform integrability, Image measures and distributions, Generating classes of sets and functions,
Chapter III is about Desnsities and Derivatives:
Densities and absolute continuity, The Lebesgue decomposition, Distances and affinities between measures, Classical concepts of absolute continuity, Vitali covering lemma, etc.
Chapter IV Product Spaces and Independence: Independence of sigma-fields, Construction of measures on product spaces, Product measures, Infinite product spaces etc.
there are 351 pages and chapters 1-4 only go to page 108. The rest of the book includes massive ammounts of data on: Conditional distributions, Integration and disintregration, Conditional densities, Invariance, Kolmogorov's abstact conditional expectation, Sufficiency. Chapter VI is all about Martingales; Stopping times, Convergence, Krickeberg decomposition, Uniform integrability, Reversed Martingales, Symmetry, Exchangeability, Lindeberg's method, Multivariate limit theorems, Fourier transforms, Martingale central limit theorem, the Levy and Cramer-Wold theorems, Brownian motion etc.
Also by Chapter 10 you get into: Representations and Couplings;
Strassen's theorem, the Yurinskii coupling, Quantile coupling of Binomials with normals, Haar coupling etc. Chapters 11 and 12: Exponential Tails and and the Law of the Iterated Logarithm; Identically distributed summands, Multivariate Normal Distributions; Ferniques inequality, and it's proof, Gaussian isoperimetric inequality, and it's prrof. The extensive Appendices include topics: Measures and inner measures, Tightness, Countable additivity, Extension to the ^c-closure, Integral representations, Hilbert Spaces; Orthogonal projections, Orthonormal bases, Series expansions, Convexity; Convex sets and functions, their Integral representations, their relative interiors, Quantile coupling of the Binomial with the normal, Martingales in Ccontinuous Time; using filtrations, Brownian filtrations, supermartingales etc. Disintegration of Measures; Representations of measures on product spaces, disintegrations with respect to a measurable map etc.
Most of the material covered has previously only been available in French contexts and we are lucky to have now in english, available to Phd's in the United States and written at the Graduate level for students worldwide.
12 of 12 people found the following review helpful
5.0 out of 5 stars
Excellent and idiosyncratic
May 10 2005
By Giuseppe A. Paleologo
- Published on Amazon.com
Format:Paperback
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.
12 of 12 people found the following review helpful
5.0 out of 5 stars
Nice Book
Oct 19 2003
By ktrmes
- Published on Amazon.com
Format:Paperback
I first saw this book online and decided that I would buy it when it became available. It is a very nice book -- one of those rare math books with material so well arranged that can be read almost like a novel. The material in Chapter 2 (A Modicum of Measure Theory) in particular is a fine example of this -- I had learned this material from Royden whose presentation, though providing just what a math graduate student should have, does not make the theorems part of a story in an historical context. This context, the level and pace of the presentation and the book's conscious acknowledgement of motivations make for a very readable presentation.
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4.1 out of 5 stars
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