Best book for self-study on the foundations of probability

Question

After some selection, I have three "candidates" books to purchase in order to study by myself the foundations of the theory of probability, at a level that I can define as "high undergraduate"/"low graduate". These are my candidates:

1) Probability. A.N.Shiryaev. GTM Springer-Verlag.

2) Probability and Stochastics. Erhan Çinlar. GTM Springer-Verlag.

3) Probability Theory. A. Borovkov. UT Springer-Verlag.

I have very good references from all of them. Which one do you recommend me ?

Thank you for your answers.

Answer 1

I have used the first two, but do not know the third one.

Shyriaev's book is more computational and shows the proofs for all the theorems and propositions, to all small details. Cinlar is more abstract and skips many intermediate steps, but goes into more depth in other directions. Both are good books, but I think Shiryaev is better for self-study.

However, I would look for a book with solutions to the problems if you are planning to do it all by yourself.

Good luck!

Answer 2

I would reccomend just using lecture notes available online. That way you have no risk from purchasing a book.

Some good ones: http://www.math.nyu.edu/faculty/varadhan/limittheorems.html

http://www-stat.stanford.edu/~adembo/stat-310a/lnotes.pdf

https://ocw.mit.edu/courses/mathematics/18-175-theory-of-probability-spring-2014/lecture-slides