There are many, many excellent texts for a graduate level course on probability. Among them are books by Resnick (A Probability Path), Gut (Probability: A Graduate Course), Pollard (A User’s Guide to Measure Theoretic Probability), Williams (Probability With Martingales), Chung (A Course In Probability Theory), and enough others to fill a shelf of my bookcase.
Two of the most popular texts are Billingsley’s Probability and Measure and Durrett’s Probability: Theory and Examples, and choosing between them is (in my opinion) a matter of taste. It is perhaps a comment on the learning process that the official text when I took this course was Billingsley, but I preferred Durrett. A few years ago, while chatting at a conference with a colleague, he mentioned that the official text when he did the course was Durrett, but that he preferred Billingsley!
I used the first edition, which was quite terse, and had many typos. Some Amazon.com reviewers also complained of the quality of the index. The book is still terse, but slightly less so — just a few more words explaining tricky parts makes this a much friendlier text than before. There are still typos, but fewer than before. Most of the ones that remain are easily avoided or pointed out by an instructor. A few require more effort, for example on page 50 the symbol ˜ is used, but only defined on page 81. I checked one of the complaints about the index, concerning the definition of Poisson distribution and it is now easy to find from the index.
These changes do not dull Durret’s wit. It is hard to believe but there were several times while reviewing the book that I laughed out loud, for example when looking for the non-existent table of the normal distribution at the back of the book.
Note that the title is Probability: Theory and Examples and that is exactly what the book contains. The theory is well developed and followed by nice examples, and then very interesting (and challenging) exercises. The examples are not fully developed applications, but rather crisp examples that illustrate the preceding theory. And yes, there are times when an exercise requires more than the immediately preceding examples and theorems. You may have to think more, and the solution may involve solutions of previous exercises — but this is what real math research is like. You never know where the answer will come from.
The book has been slightly reorganized from previous editions, with part of the measure theory appendix becoming chapter 1, and the remainder still being in the appendix.
This is not a book for beginners but a rigorous text, aimed at people who want to use probability in a profound way. As such, it makes serious demands on the reader, but yields corresponding benefits to those who persevere.
And I still like it better than Billingsley’s book.
Peter Rabinovitch is a Systems Architect at Research in Motion, and a PhD student in probability. He likes spicy food and Disney World and hates losing money at casinos.
1. Measure theory; 2. Laws of large numbers; 3. Central limit theorems; 4. Random walks; 5. Martingales; 6. Markov chains; 7. Ergodic theorems; 8. Brownian motion; Appendix A. Measure theory details.