I am reading Kai Lai Chung's A Course in Probability Theory. I understood the concept of vague convergence of a sequence of probability measures in the book. But it seems to me the book uses the concept of vague convergence of a sequence of distribution functions without defining it first. But I could miss the definition, and if you can point me where the book defines it, it will be nice.
So I was wondering
- how the vague convergence of a sequence of distribution functions is defined?
- how it is related to the vague convergence of the corresponding probability measures?
- how it is related to the point-wise convergence of the distribution functions?
- ADDED: for a sequence of (sub)probability measures or a sequence of distribution functions, how weak convergence is defined? How it and vague convergence are different from each other?
Thanks and regards!