In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ... The terms almost certainly (a.c.) and almost always (a.a.) are also used.
What is the difference between convergence in probability and almost surely?
The sequence of random variables will equal the target value asymptotically but you cannot predict at what point it will happen. Almost sure convergence is a stronger condition on the behavior of a sequence of random variables because it states that "something will definitely happen" (we just don't know when).
What does it mean to converge almost surely?
Almost sure representation.
When Xnconverges almost completely towards X then it also converges almost surely to X. In other words, if Xn converges in probability to X sufficiently quickly (i.e. the above sequence of tail probabilities is summable for all ε > 0), then Xn also converges almost surely to X.
