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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.

Richard Hardy
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HBat
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4 Answers4

31

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

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

Matt
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23

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!

Cliff AB
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  • *"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. – user541686 Mar 02 '17 at 19:31
  • @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 $X \ne 0.5$. – Cliff AB Mar 02 '17 at 19:41
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    Yup... might want to modify your answer accordingly... – user541686 Mar 02 '17 at 19:49
  • @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. – David Richerby Mar 03 '17 at 10:33
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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 $$Pr(\lim_{n\to\infty}{X_n}=X)=1$$

Daniel Salgado Olvera
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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.

Shlomi A
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