Advanced stochastic process book (a bit flavor from real analysis)

Question

I am looking for the book about advanced stochastic process.

It may cover the following content:

1. Stochastic matrices. Ex: $A(k)$, where $k$ is the time index.
2. Stochastic process in space (not just in time).
3. Contents from real analysis (Ex: application of monotone convergence theorem, Fatou' Lemma, increasing sequence of random variables).
4. Miscellaneous topics (such as supermartingale convergence theorem, or convex function discusstion).

I am not sure my description is accurate or not, but I will fix it if possible.