Almost sure convergence

Definition

Let $X, X_1, X_2, \dots$ be real random variables on one probability space $(\Omega, \mathcal{F}, P)$. The sequence $(X_n)$ converges almost surely to $X$ if $$P\left(\left\{\omega \in \Omega : \lim_{n \to \infty} X_n(\omega) = X(\omega)\right\}\right) = 1.$$ One writes $X_n \to X$ a.s. or $X_n \xrightarrow{\text{a.s.}} X$.

Remarks

The set in the definition is an event, because the $X_n$ and $X$ are measurable. An equivalent condition: for each $\varepsilon > 0$, $P(|X_n - X| > \varepsilon \text{ for infinitely many } n) = 0$. The limit is unique up to equality almost surely. The strong law of large numbers is a result about almost sure convergence.

Explanations

L1 human

Almost sure convergence means that, apart from a set of outcomes of probability $0$, each run of the experiment gives a convergent sequence of numbers.