Almost surely
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In probability theory, the phrase almost surely is a concise, precise way to state except on a set or event of probability measure zero. Heuristically, almost sure events are those which have zero probability of not occurring though it is still possible that they might not occur. The concept is often encountered in questions that involve infinite time, infinite-dimensional spaces such as function spaces, or infinitesimals.
The locution almost surely is to probability theory as almost everywhere is to measure theory. Formally, it is equivalent to "with probability 1". For example, imagine throwing a dart at the unit square (i.e., selecting a random point within the square); the probability that the dart lands in any subregion of the square is the area of that subregion. The area of the diagonal of the square is zero, so the probability that the dart lands exactly on the diagonal is zero; however, the diagonal is not the empty set; a point on the diagonal is no less probable than is any other point at which the dart could land. One says that the dart will almost surely not land on the diagonal. In other words, an event is "almost sure" if the probability of its complement is zero, even though the complement may not be empty. (Note that the setting described is very idealized also in that it depends entirely on an idealized notion of space and spatial measurement: the width of the diagonal is assumed to be zero, and the width of the tip of the dart is likewise assumed to be zero, and the precision of determining where exactly the dart hit is assumed to be arbitrarily precise, i.e. infinite or zero; of course this is not the situation anybody could realize with a physical dart thrown at a physical square with a physically materialized diagonal and trying to physically measure exactly what happened. As soon as such physical inevitabilities have to be taken into account, the probability of hitting the diagonal, although it may still be "very small", grows above zero accordingly, and the diagonal will not be avoided almost surely any longer.)
To state it more mathematically, the probability of an event being 0 means that as the number of trials tends to infinity, the limit superior of the ratio of successes to trials is zero. Thus, a dart can hit the diagonal many times and still the probability of hitting the diagonal is zero, provided that as the number of trials (throws of the dart) becomes larger and larger, the mean interval between hits becomes even larger yet, and in such a way that the ratio of hits to throws has limit superior of zero. Similarly, for an event to have probability 1, it is only necessary that the limit inferior of the ratio of successes to trials, as the number of trials tends to infinity, be 1. This means that occasional failures may happen, but as the number of trials increases, the intervals between failures must increase sufficiently fast that the limit inferior just mentioned is 1.
Instead of infinitesimal space, one may consider infinite time. Suppose that a coin is flipped again and again.A sequence heads, heads, heads, ..., ad infinitum, without ever coming up tails, is possible in some sense -- it does not violate any physical or mathematical laws to suppose that tails never appears -- but it is very, very improbable. In fact, such a sequence has probability zero; however, any real sequence must have a finite length, and any possible realization has probability greater than zero. The difficulty comes when the sequence is hypothetically extended to infinite length. It is common to talk about almost sure convergence or divergence of random variables.
An example of the fine distinction between 'sure' and 'almost sure' can be found in the difference between constant and almost surely constant random variables. In measure theoretic probability theory these two types of random variable are not identical, but for practical purposes they are equivalent, since if a constant random variable X and an almost surely constant random variable Y represent the same constant c, then they share the same cumulative distribution functions.
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