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A User's Guide to Measure Theoretic Probability (Cambridge Series in Statistical and Probabilistic Mathematics) [Paperback]

David Pollard
1.0 out of 5 stars  See all reviews (1 customer review)
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Book Description

10 Dec 2001 Cambridge Series in Statistical and Probabilistic Mathematics (Book 8)
Rigorous probabilistic arguments, built on the foundation of measure theory introduced eighty years ago by Kolmogorov, have invaded many fields. 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 2002 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.

Product details

  • Paperback: 366 pages
  • Publisher: Cambridge University Press (10 Dec 2001)
  • Language: English
  • ISBN-10: 0521002893
  • ISBN-13: 978-0521002899
  • Product Dimensions: 2.1 x 17.9 x 25.1 cm
  • Average Customer Review: 1.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Bestsellers Rank: 1,175,534 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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Review

'A really useful book …'. EMS Newsletter

Book Description

This 2002 book offers a rigorous probability course for a mixed audience - statisticians, biostatisticians, mathematicians, economists, and students of finance - at the advanced undergraduate/introductory graduate level, without measure theory as a prerequisite. It covers the basic topics of independence, conditioning, martingales, convergence in distribution, and Fourier transforms plus more advanced topics.

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2 of 3 people found the following review helpful
1.0 out of 5 stars disappointing book of little use 7 May 2009
Format:Paperback
This is not a book, but a syllabus in the worst sense of the word.
It consists of theorems and definitions, with an occasional example thrown in.If you expect a good exposition of measure theoretic issues, you will be disappointed. The reader is mainly referred to other books.
Several key results are hidden in exercises. There are no solutions or hints for exercises, so it is ill suited for self study, or as a companion or back up book for your university study.
I bought the book on the fact that it is part from an excellent series.
(the books on Markov Chain's, Bayesian Methods and the Bootstrap are all very good).
My advice: stay away from this book, there are far better books on this subject.
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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! 26 Sep 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
[...]
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 Nice Book 20 Oct 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.
12 of 12 people found the following review helpful
5.0 out of 5 stars Excellent and idiosyncratic 11 May 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.
4 of 4 people found the following review helpful
5.0 out of 5 stars fantastic but very difficult 18 Feb 2005
By a student - Published on Amazon.com
Format:Paperback
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.
4 of 4 people found the following review helpful
5.0 out of 5 stars interesting perspective 22 Sep 2007
By Moral Lizard - Published on Amazon.com
Format:Paperback|Verified Purchase
I spent long hours learning probability theory from Billingsley's book, 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. After having finished the book, I wend ahead to buy a copy.

Contrary to an earlier reviewer, I appreciate very much the same symbol for probability and expectation, which allows a quite natural and unified treatment of the two important concepts. It brings great clarity. Also, de Finetti's notation, once introduced, seems only natural.

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!
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