Please recommend a nice and concise math book on probability theory.

My intention is neither to learn basic probability concepts, nor to learn applications of the theory. My background is at the graduate level of having completed all engineering courses in probability/statistics -- mostly oriented toward the applications without much emphasis on mathematical rigor.

Now I am very interested in learning the core logic and mathematical framework of probability theory, as a math branch. More specifically, I would like to learn answers to the following questions:

(1) What are the necessary axioms from which we can build probability theory?

(2) What are the core theorems and results in the mathematical theory of probability?

(3) What are the derived rules for reasoning/inference, based on the theorems/results in probability theory?

So I am seeking a book that covers the "heart" of mathematical probability theory -- not needing much on applications, or discussion on extended topics.

I would like to appreciate your patience for reading my post and any informative responses.

Regards, user36125

• When you say not much need for applications, Probability is a subject that is very particularly learnt by working on examples. The book I've recommended has many fully worked examples which go a long way to explaining the concepts involved. – Geoff Pointer Jan 28 '14 at 21:56
• Probability Theory by Varadhan is short, affordable, and covers the core results. It is someone terse though. – Michael Greinecker Jan 28 '14 at 22:12
• Hi guys, thank you all for your time and response. I particularly like Probability Theory by Varadhan (@ MichaelGreinecker) and Probability with Martingales by David Williams (@user901823). Both of them seem to be terse, yet have explained rationales clearly. – user36125 Jan 31 '14 at 7:54
• @GeoffPointer - Thanks so much! Elementary Probability by David Stirzaker appears to be an excellent introductory book with lots of examples. But this is not the type of book I am looking for, since my intention is mostly to learn the pure math foundations of probability. I have a few pure math questions in my mind that motivated this learning, though I am not a mathematician :) – user36125 Jan 31 '14 at 8:06

Williams, Probability with Martingales

I personally liked Davidson's Stochastic Limit Theory. It rigorously introduces measure-theoretic probability, and then proves the key convergence results. Towards the end it rigorously develops continuous-time stochastic processes.

You mention inference, so if you are particularly interested in the statistics rather than just probability, I would also recommend Schervich's Theory of Statistics.

If you'd like to see a mathematical development, but you find you don't care for measure theory, then Lehman's Elements of Large-Sample Theory is pretty good.

A good collection of books on probability theory and statistics is listed here. It is a good collection in the sense it has all ranges of books, from core concepts to lighter reading. Additionally, there is no better way to learn it than to learn a programming language like R and a scripting language like Perl/Python. Hope that helps.

Get a copy of Elementary Probability by David Stirzaker from the library and take it from there. If you like it then buy a copy. I bought this book on a whim some years after majoring in pure maths and it rekindled my interest in probability.

Brief Review: It starts right from the beginning of Probability so very little previous knowledge of the subject is required. It has a chapter dedicated to Counting which is a particularly favourite topic of mine. It introduces pretty much every branch of Probability and gives ample opportunity for you to decide whether or not you'd want to take any particular area or areas any further.

Billingsley's Probability and Measure covers many interesting topics and has excellent problems. It has an unusual approach: building up the mathematical structure of subject not at once but in a number of iterative steps of increasing complexity. That takes up time but I have still found this repetition helpful in consolidating my understanding.

Being very comfortable with real analysis at the level of Rudin's Principles is an essential prerequisite for this book.

If you are buying this avoid the "Anniversary Edition" which is littered with typos. Try to get a copy of the 3rd edition instead.

• I agree. Billingsley's Probability and Measure was very helpful f.e. for my bachelor thesis. – math12 Jan 29 '14 at 14:39