- Series: Probability: Theory & Examples
- Hardcover: 512 pages
- Publisher: Duxbury Press; 3 edition (March 16, 2004)
- Language: English
- ISBN-10: 9780534424411
- ISBN-13: 978-0534424411
- ASIN: 0534424414
- Product Dimensions: 6.5 x 1 x 9.2 inches
- Shipping Weight: 1.8 pounds (View shipping rates and policies)
- Average Customer Review: 38 customer reviews
- Amazon Best Sellers Rank: #1,022,149 in Books (See Top 100 in Books)
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Probability: Theory and Examples (Probability: Theory & Examples) 3rd Edition
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More precisely, this book is comprehensive in terms of SKILLS that you will need to pass prelims/do research.
Though, it is not very good to 'tell the story of probability'. For instance, section 5.4 is called 'Doob’s Inequality, Convergence in Lp'. If you are new to this area, you might wonder why we should study convergence in Lp of martingales. Severe lack of connections between theorems/sections. If you are an undergrad trying to find your interest / grad student in other discipline but want to use some grad probability, for instance, Maybe 'A Probability Path' by Resnick, or 'A First Look at Rigorous Probability' by Rosenthal are better for this purpose.
It's a good book, but not for everyone.
Being done with the class, I now find the book an excellent reference - it is a very concise presentation of the material.
I recommend this book if you're a student in a graduate-level class or if you're already familiar with the material. But if you want to teach yourself, this book is too difficult a starting point.
First of all, I admit that this isn't a book I would want to first learn from (I did learn from it, so I would know). When I was learning probability theory, I already knew measure theory, and so I wanted a book that would actually use measure theory freely. There are many important theorems, such as the central limit theorem, which demand that the reviewer use measure theory. It is the avoidance of measure theory which makes most elementary books on probability theory uninteresting. On the other hand, most measure-theoretic probability theory books assume that you've learned from an elementary book and so they don't waste time on developing intuition. I've found that most books, Billingsley's included, simply treat probability theory as an extension of measure theory. This is not what I wanted, and I was led to this book.
Probability theory is a field with one foot in examples and applications and the other in theory. The thing that this book does better than others, except perhaps for the beautiful, but infinitely long Feller, is that it pays homage to the applications of probability theory. This should be expected, judging from the title. Every page of this book has an example. Every single theorem is used in an interesting example, and there are tons of exercises asking the reader to use the theorems and prove alternative theorems. A student would not leave with a healthy perspective of probability theory if all they learned from were, say, Varadhan's notes or Stroock's analytic book of death. They would know a whole lot of extra measure theory without having any idea of what problems this theory was designed to solve. A student working with Durrett's book will have seen plenty of examples and worked out a huge number of problems themselves. Hence, they will leave this book with a deeper understanding and a more balanced view of probability theory. It is these applications which attracts so many researchers to this book and make them impose it on their students.
Now, this book is massive. It sits at a relatively slim 400 pages, but just about all of basic probability theory is in here: sums of independent random variables, basic limit theorems, martingales, markov chains, brownian motion, etc... There is nothing that a student could wish for (except coupling) which
they won't find in here. As always, trying to learn from massive, sprawling books is a challenge. This expansive coverage makes the book great to have around as a reference and second textbook. There are also tons of tricks that he teaches which are really useful and aren't found easily in other books.
Now, there are many complaints about the readability of this book. In the fourth edition, Durrett switched over to LaTeX (as opposed to just TeX I believe) and so the cross references and index have all be corrected. I found the typos in this book to still be too large in number, but they are almost universally trivial. None of the typos confused me. I will blame Durrett for being careless, but not an incompetent expositor. There is one wrong proof somewhere in here that my professor mentioned, so that's unfortunate. Also, when I was first learning probability theory, I found this book difficult to read. However, now that I know a good amount I find this book to be perfectly readable. I suppose it's just best as a second book.
Overall, this book does have some flaws (hence, the subtraction of a star), but students will leave with a healthy perspective. This is why this is the best measure-theoretic probability book.
First, apparently this book contains a lot of examples. Important examples are necessary for building intuition. However, many of the examples in this book are not essential. So if you work though the examples, it's likely you will not be able to follow the flow of theorems uninterruptedly (being able to see structure of the theory is very important for me); if you skip some of the examples, it's bad because they are referenced in the some of the proofs of theorems.
Second, this book does not differentiate propositions, lemmas, corollaries, and theorems. The names of some theorems help, but they are insufficient for me to tell which ones are the main nontrivial results, i.e. the cornerstones of the theory.
Third, some of the important intuitions pop up in the middle of nowhere. For example, an important intuition (or conceptional foundation) is that sigma algebras encode information, and measurable functions (random variables) represent outcomes that are knowable based on the available information. Nowhere in this book I see this, but when discussing conditional expectation, this intuition suddenly becomes indispensable.