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I am looking for a book (English only) that I can treat as a reference text (more colloquially as a bible) about probability and is as complete - with respect to an undergraduate/graduate education in Mathematics - as possible. What I mean by that is that the book should contain and rigorously address the following topics:

  • Measure Theory (As a mathematical foundation for probability)
    • It is of course fine if this theory is addressed with an emphasis on probability and not only for the sake of mathematical measure theory, although the latter would be great too.
  • Introduction to Probability, i.e. the most common theory a student is exposed to when taking a first course in theoretic Probability. For example: distributions, expected value, modes of convergence, Borel Cantelli Lemmas, LLN, CLT, Gaussian Random Vectors
  • More advanced topics such as: Conditional Expectation (defined through sigma-Algebras), Martingales, Markov Processes, Brownian Motion

I want it to be one book so I can carry a physical copy of it with me and work through the material in my spare time.

Examples:

  1. The book by Jean-Francois Le Gall which can be found here: https://www.math.u-psud.fr/~jflegall/IPPA2.pdf but (unfortunately for me) is written in French.
  2. Rick Durrett's book on Probability which can be found here https://services.math.duke.edu/~rtd/PTE/pte.html - the critique available for this book seems a bit mixed, uncertain about how to weigh that.

I am well aware that it's not easy to meet all of the above criteria simultaneously, but I would be grateful for any recommendation.

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  • $\begingroup$ Many people like this book amazon.com/gp/product/0521406056/… $\endgroup$
    – meh
    Commented Oct 10, 2016 at 15:29
  • $\begingroup$ I dont think there's one book that covers it all, and also readable $\endgroup$
    – Aksakal
    Commented Oct 10, 2016 at 18:18
  • $\begingroup$ My favourite introduction to measure-based probability, for readability and choice of topics, is B. Candelpergher, Théorie des probabilités. Une introduction élémentaire. Unfortunately, it's again in French. Are you sure that you don't want to learn a bit of French? ;-) $\endgroup$
    – Massimo Ortolano
    Commented Oct 10, 2016 at 18:46
  • $\begingroup$ @MassimoOrtolano, you got me there! I really should learn French. $\endgroup$
    – Spaced
    Commented Oct 10, 2016 at 18:54
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    $\begingroup$ You should never treat any text "as a bible". No text deserves an assumption of inerrancy, for starters. Books are written by humans and humans make errors. Many texts are useful, but even the good ones have their bad points. Secondly, I think it's a mistake to rely only on one text, even if it's a really good one. It's important to read more widely if one is to gain a robust understanding of many issues. $\endgroup$
    – Glen_b
    Commented Oct 10, 2016 at 23:36

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I suggest you a couple of books that I admit I never had the occasion to study. These would have been my reference if I specialized in probability:

  1. Ash, Dade - "Probability and Measure Theory"
  2. Billingsley - "Probability and Measure"

I think (2) is more popular.

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    $\begingroup$ If you have never studied them perhaps you could expand on why you recommend them? $\endgroup$
    – mdewey
    Commented Oct 10, 2016 at 15:58
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    $\begingroup$ Oh, yes, time ago i thought it was important for me to learn well the mathematical fundations of Probability theory. So I started, like you, to look around for a reference book to use as a "Bible". I looked into many books and these 2 are the ones which I most liked. So, I suggest you give them I look. You can find the pdf around. Read some parts that you more or less know already and build your opinion. Good luck! $\endgroup$
    – Nicola Mingotti
    Commented Oct 10, 2016 at 16:09
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    $\begingroup$ Billingsley's book is a masterpiece-- I wish every mathematics texrbook were written by him. $\endgroup$
    – tooAnnoying
    Commented Mar 19, 2021 at 16:42
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Foundations of Modern Probability by Olav Kallenberg meets all your criteria. It is quite concise and mathematically rigorous and as one reviewer puts it "without any non-mathematical distractions".

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I think Amir Dembo's notes are pretty stellar. He updates them each time he teaches the course, but even then they have really good proofs and exercises. He also has notes on stochastic processes. William's Probability with Martingales is also good... but only the parts on martingales are good. Durrett's book is decent.

I am of the persuasion that stochastic processes should be done in depth as its own course, and for the Oksendal "Stochastic Differential Equations" is easier and more insightful than Karatzas "Brownian Motion and Stochastic Calculus" which is tougher but more thorough, together making a good combo. I came out the other side still wanting to learn Malliavin calculus (still haven't gotten around to it) but feeling ready to do so.

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    $\begingroup$ The work by Amir Dembo looks very promising! I am slightly intimidated by the preface where it says that the text is intended for a year long PhD level course in Probability but it looks very well written. Thanks a lot for sharing this with me. $\endgroup$
    – Spaced
    Commented Oct 10, 2016 at 19:26
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I know these are no books, but nonetheless I think these materials are quite useful:

At MIT they offer various courses for free. Some of these courses might also contain books.

Overview of free probability and statistics courses at MIT

This is the best online course of advanced Theory of probability. It is made by Scott Sheffield, who is the most famous probability professor at MIT.

Theory of probability

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Weiss, A Course in Probability covers at least nearly all of these topics. It's also very readable.

There's a strange problem with probability and stats textbooks where the notation of explanation is exceptionally shoddy and non-rigorous. This book usually doesn't suffer from that deficiency. But occasionally it punts on topics that would require a familiarity with measure-theoretic concepts.

http://www.amazon.com/Course-Probability-Neil-A-Weiss/dp/0201774712

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  • $\begingroup$ This has rather poor reviews compared to others. $\endgroup$
    – JohnK
    Commented Mar 27, 2017 at 10:23
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I wholeheartedly recommend Parzen's Modern Probability Theory and its Applications. This is a classic written by someone who has made enormous contributions to statistics, in e.g. non-parametric density estimation, and is very easy to read. Moreover, it covers almost all of your required topics (I only can't remember if more advanced forms of the CLT are discussed) and presents a lot of motivating examples. The cherry on top is that since this is an old book, you can buy it for 3 dollars on Amazon or even get it for free on the internet.

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Probability Theory: The Logic of Science by E. T. Jaynes

Its a classic for a reason. I only can recommend it since it takes a more practical approach. Also contains a bunch of exercises. PDF of older versions should be available (amazon link)

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    $\begingroup$ While this might be good for opinions and ideas, it does not cover the syllabus given! $\endgroup$
    – kjetil b halvorsen ♦
    Commented Mar 27, 2017 at 11:34

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