What is the meaning of a.s.? What is the meaning of a.s. behind a limit formula (I found this in a paper about stochastic processes) , or sometimes P-a.s.?

It means almost surely. P-a.s means almost surely with respect to probability measure P. For more details wiki out "almost sure convergence".
Let me give some insights: When working with convergence of sequences of random variables(in general stochastic processes), it is not necessary for convergence to happen for all $w \in \Omega$, where $\Omega$ is the sample space. Instead it is fine if the set where it doesn't converge happens over a set with measure 0, since most of the results go through. If you take a measure theory course, you will be able to appreciate this even more.
