My background is mostly related to analysis, but I've recently run into a problem for which the central limit theorem naturally arises. I find that I need to understand the delicate issues like rate of convergence, a la the Berry-Esseen theorem, but my background in probability is insufficient to understand the language in which such theorems are formulated. To remedy this, I want to get a book to teach myself probability.
I would like recommendations for probability books which satisfy the following:
- Fully rigorous.
- Good description of the central limit theorem, Berry-Esseen theorem, and related rate-of-convergence issues.
- Preferably concise, but clear and well written. (I am familiar with measure theory, etc.)
I would highly recommend Kallenberg's textbook https://www.springer.com/gp/book/9780387953137 which covers the central limit theorem but not the rate of convergence. For the latter, see chapter 3 of https://services.math.duke.edu/~rtd/PTE/pte.html