I was reading an article and I saw the following sentence:

For a given martingale, if it has an upper or a lower bound, then the martingale must converge (a.s.). Since the likelihood is always nonnegative, 0 is a lower bound.

What does "a.s." stand for? Is it a common usage? My guess is "asymptotically" but I'd like to verify.

  • 19
    $\begingroup$ It stands for almost surely $\endgroup$
    – user33484
    Commented Mar 2, 2017 at 16:24
  • 2
    $\begingroup$ @user33484 Please don't post answers as comments. $\endgroup$ Commented Mar 3, 2017 at 10:32
  • $\begingroup$ Yes it is common usage. $\endgroup$
    – Augustin
    Commented Mar 3, 2017 at 11:45
  • $\begingroup$ @user33484 yeah, you basically lost 200-300 rep because it was a comment :P. Opportunity cost of 0. $\endgroup$
    – Nick T
    Commented Mar 3, 2017 at 18:26
  • $\begingroup$ a.s. stands for almost surely a.s. $\endgroup$ Commented Mar 5, 2017 at 21:34

4 Answers 4


It stands for "almost surely," i.e. the probability of this occurring is 1.

See: https://en.wikipedia.org/wiki/Almost_surely


As noted by @Matt, a.s. stands for "almost surely", or with probability 1.

Why the "almost" in "almost surely"? Because just because something happens "almost surely" does not mean it must happen. For example, suppose $X \sim$ Uniform(0,1). What's $P(X = 0.5)$? Well, since $X$ is a continuous random variable, $P(X = $ any finite set of values) = 0. Therefore, $X$ is almost surely not equal to 0.5. But that's not to say $X$ cannot be equal to 0.5!

  • $\begingroup$ "Just because something does not happen almost surely does not mean it cannot happen"... well obviously. A fair coin doesn't come up heads almost surely but it can still come up heads. I think you meant to say something else. $\endgroup$
    – user541686
    Commented Mar 2, 2017 at 19:31
  • $\begingroup$ @Mehrdad: Ah, there's some English ambiguity here. A less ambiguous statement: Just because $A$ happens a.s. does not mean it's impossible for $A$ not to happen. So in my example, $A$ is $X \ne 0.5$. $\endgroup$
    – Cliff AB
    Commented Mar 2, 2017 at 19:41
  • 1
    $\begingroup$ Yup... might want to modify your answer accordingly... $\endgroup$
    – user541686
    Commented Mar 2, 2017 at 19:49
  • $\begingroup$ @Mehrdad Yeah, the intended parsing was "Just because (something does not happen) almost surely"; "Just because, almost surely, something does not happen" would have been clearer. $\endgroup$ Commented Mar 3, 2017 at 10:33

As mentioned above, a. s. stands for almost shurely, but in this case they are talking about almost shurely convergence. From the Wikipedia,

To say that the sequence $X_n$ converges almost surely or almost everywhere or with probability 1 or strongly towards $X$ means that $$Pr(\lim_{n\to\infty}{X_n}=X)=1$$


As already noted by others, "a.s." stands for "almost surely". The wikipedia article quoted by @Matt is a good start for almost surely and its synonyms.

There is however a subtle distinction between almost surely (or with probability 1) to always [resp., between with probability zero to never].

Imagine an infinite series of i.i.d. random variables which are head a.s. (=with probability 1), tail with probability zero. It is possible in such an infinite series to have a finite number of tails although the probability for tail is 0, as the empirical distribution of the series remains 1-0 (only a finite number of instances out of infinitely many). On the other hand, when one says that the series is always head one means that not even a single tail occurs in the series.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.