Almost surely

In probability theory, one says that an event happens almost surely (sometimes abbreviated as a.s.) if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory. While there is no difference between almost surely and surely (that is, entirely certain to happen) in many basic probability experiments, the distinction is important in more complex cases relating to some sort of infinity. For instance, the term is often encountered in questions that involve infinite time, regularity properties or infinite-dimensional spaces such as function spaces. Basic examples of use include the law of large numbers (strong form) or continuity of Brownian paths.

Almost never describes the opposite of almost surely; an event which happens with probability zero happens almost never.^[1]

Contents 1 Formal definition 2 "Almost sure" versus "sure" 2.1 Throwing a dart 2.2 Tossing a coin 3 Asymptotically almost surely 4 See also 5 Notes 6 References

Formal definition

Let (Ω, F, P) be a probability space. One says that an event E in F happens almost surely if P(E) = 1. Equivalently, we can say an event E happens almost surely if the probability of E not occurring is zero.

An alternative definition from a measure theoretic-perspective is that (since P is a measure over Ω) E happens almost surely if E = Ω almost everywhere.

"Almost sure" versus "sure"

The difference between an event being almost sure and sure is the same as the subtle difference between something happening with probability 1 and happening always.

If an event is sure, then it will always happen, and no outcome not in this event can possibly occur. If an event is almost sure, then outcomes not in this event are theoretically possible; however, the probability of such an outcome occurring is smaller than any fixed positive probability, and therefore must be 0. Thus, one cannot definitively say that these outcomes will never occur, but can for most purposes assume this to be true.

Throwing a dart

For example, imagine throwing a dart at a unit square wherein the dart will impact exactly one point, and imagine that this square is the only thing in the universe. There is physically nowhere else for the dart to land. Then, the event that "the dart hits the square" is a sure event. No other alternative is imaginable.

Next, consider the event that "the dart hits the diagonal of the unit square exactly". The probability that the dart lands on any subregion of the square is proportional to the area of that subregion. But, since the area of the diagonal of the square is zero, the probability that the dart lands exactly on the diagonal is zero. So, the dart will almost surely not land on the diagonal. Nonetheless the set of points on the diagonal is not empty and a point on the diagonal is no less possible than any other point, therefore theoretically it is possible that the dart actually hits the diagonal.

The same may be said of any point on the square. Any such point P will contain zero area and so will have zero probability of being hit by the dart. However, the dart clearly must hit the square somewhere. Therefore, in this case, it is not only possible or imaginable that an event with zero probability will occur; one must occur. Thus, we would not want to say we were certain that a given event would not occur, but rather almost certain.

Another example of the "dart throwing" kind actually contradicting the above. When throwing a dart, we may be interested in one coordinate only. Let's suppose we are interested in the horizontal one. Now, the event, that we obtain an irrational number has probability one, i.e. it is almost sure.

Tossing a coin

Suppose that an "ideal" (edgeless) fair coin is flipped again and again. A coin has two sides, head and tail, and therefore the event that "head or tail is flipped" is a sure event. There can be no other result from such a coin.

The infinite sequence of all heads (H-H-H-H-H-H-...), ad infinitum, is possible in some sense (it does not violate any physical or mathematical laws to suppose that tails never appear), but it is very, very improbable. In fact, the probability of tail never being flipped in an infinite series is zero. Thus, though we cannot definitely say tail will be flipped at least once, we can say there will almost surely be at least one tail in an infinite sequence of flips. (Note that given the statements made in this paragraph, any predefined ordering would have zero-probability in an infinite series. This makes sense because there are an infinite number of possibilities and .)

However, if instead of an infinite number of flips we stop flipping after some finite time, say a million flips, then the all-heads sequence has non-zero probability. The all-heads sequence has probability 2^−1,000,000, thus the probability of getting at least one tail is 1 − 2^−1,000,000 < 1, and the event is no longer almost sure.

Asymptotically almost surely

In asymptotic analysis, one says that a property holds asymptotically almost surely (a.a.s.) if, over a sequence of sets, the probability converges to 1. For instance, a large number is asymptotically almost surely composite, by the prime number theorem; and in random graph theory, the statement "G(n,p_n) is connected" (where G(n,p) denotes the graphs on n vertices with edge probability p) is true a.a.s when p_n > for any ε > 0.^[2]

In number theory this is referred to as "almost all", as in "almost all numbers are composite". Similarly, in graph theory, this is sometimes referred to as "almost surely".^[3]

See also

• Convergence of random variables, for "almost sure convergence"
• Degenerate distribution, for "almost surely constant"
• Almost everywhere, the corresponding concept in measure theory
• Infinite monkey theorem, a theorem using the aforementioned terms.

Notes

References

• Rogers, L. C. G.; Williams, David (2000). Diffusions, Markov Processes, and Martingales. 1. Cambridge University Press. 
• Williams, David (1991). Probability with Martingales. Cambridge University Press.