I have checked several answers that request graduate-level probability theory textbooks, but I have wanted to receive advice that fits my particular needs.
My background is in mathematics and I am currently pursuing Ph.D. degree in statistics, but I am still more geared toward mathematics. In my first-year Ph.D. course for probability theory, we used Durrett's Probability Theory and Examples, but I found that the book is too terse for a first reading. I had no problem reading Terrace Tao's Introduction to Measure Theory and Folland's Real Analysis in my real analysis course, but the proofs in Durrett's book were often incomplete and most of the exercises were too hard to solve for someone who is studying the topic for the first time. Therefore, I am planning to use another textbook to supplement Durrett's, and I wanted to ask you advice on which books would be the best choice for me
I wish the book is suitable for self-study, that is, it should have rather complete proofs to show each step especially in the earlier chapters. Since I have background in mathematics, I am looking for a book that is still rigorous so that it does not omit significant amount of proofs.
Some of the alternatives I have found are Erhan Çinlar's Probability and Stochastics and Shiryaev's Probability 1, 2. I have asked to my professor about the textbook and it seems like he approves of it, but I wanted to see other options as well. I would love to hear your recommendations and your rationales behind recommending the books.
[Edited] Added Shiryaev's Probability series.
Best Answer
I highly recommend Jean-Francois Le Gall's "Brownian Motion, Martingales and Stochastic Calculus", as I have used this book for self-study myself. As the title says, it focuses mainly on stochastic calculus, but also contains some general theory of martingales and Markov processes. I believe that these topics are usually covered in advanced probability courses.
I also recommend Kallenberg's "Foundations of modern probability", which covers all the basics, but also has quite some depth, perhaps even too much depth for self-study. If Kallenberg's book is too much and you're more interested in getting an comprehensive introduction to the basics of probability and measure theory, I recommend Billingsley's "Probability and Measure".