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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.

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    $\begingroup$ Not sure if it helps, but a book on quantum mechanics I recently started introduced me to the fact that there are multiple takes on probability theory and its foundations, so to that end, you might be interested in some of the books recommended there: Probability Theory: the Logic of Science by E.T. Jaynes (unfortunately, somewhat incomplete due to his passing) and Theories of Probability: Examination of Foundations by T.L.Fine $\endgroup$
    – Yuriy S
    Dec 20, 2020 at 2:28
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    $\begingroup$ springer.com/gp/book/9780387329031 $\endgroup$
    – cqfd
    Dec 20, 2020 at 2:43
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    $\begingroup$ It's probably worth your looking at Billingsley's Probability and Measure, the third edition of which is widely regarded as a classic in the field. I haven't looked at it myself, but two other books of his, Convergence of Probability Measures, and Ergodic Theory and Information, which I own copies of, are wonderful works, demonstrating Billingsley's rare gift of making a perfectly rigorous exposition a pleasure to read. $\endgroup$
    – lonza leggiera
    Dec 20, 2020 at 3:15
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    $\begingroup$ Some reviews of Billingsley's Probability and Measure on Amazon maintain that the Anniversary Edition of $2012$, linked to above, introduced lots of errors not present in earlier editions, so you're probably better off checking out the third edition of $1995$. $\endgroup$
    – lonza leggiera
    Dec 20, 2020 at 3:16
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    $\begingroup$ I have seen early versions of Billingsley's book in his classes. I don't think you'll find clearer expositions or better sequencing of important topics in measure theoretic prob. But I have nothing to say against the 'alternative' books you've discussed w/ your adviser. // My main concern is that for your progress toward a Ph.D. in statistics, you need to adopt (at least for the duration of the program) an unambiguous and single minded approach in the subject matter that interests you. Find a thesis topic ASAP and pursue it without distraction. // For the rest of your life, do what you want. $\endgroup$
    – BruceET
    Dec 20, 2020 at 5:20

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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".

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  • $\begingroup$ Thank you for your recommendations. I think the first book lacks the usual fundamental development for probability theory, and I actually think Kallenberg's book might suit better for me. Someone pointed out that Billingsley's expositions are not particularly better than Durrett's, so I will look into the second book more. $\endgroup$
    – Taxxi
    Dec 21, 2020 at 11:09

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