Text on convergence theorems in probability theory (various modes of convergence)

Question

I need a text reviewing theorems and discussing with details ALL the types of convergence in probability theory such as almost sure convergence, convergence in probability, weak convergence, $L^p$ convergence, ... .

Answer 1

It isn't a probability theory book, but Folland's Real Analysis: Modern Techniques and Their Applications covers all of these types of convergence (note that convergence in probability is a special case of convergence in measure).

Answer 2

You should try Shiryaev's boook. You will find everything you need with a good level of details.

Probability - Shiryaev