# 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". –  Gautam Shenoy Nov 20 '12 at 14:15
@GautamShenoy: You should post that as an Answer! –  hardmath Nov 20 '12 at 14:17

## 1 Answer

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.

-
I'm a CS student with little mathematical knowledge, thank you for your advice, I will find some information about measure theory –  windy Nov 20 '12 at 14:30
Hopefully you have an analysis background. –  Gautam Shenoy Nov 20 '12 at 14:32
Then may I ask what is analysis background about? Thanks –  windy Nov 20 '12 at 14:39
By that I mean the notion of limits, continuity, differentiability, integration. Basically first 7 chapters of "Principles of mathematical analysis", by Walter Rudin. –  Gautam Shenoy Nov 20 '12 at 14:40
Thank you, this helps a lot! –  windy Nov 20 '12 at 14:43