Editing Convergence of random variables (section)

===Definition===
To say that the sequence {{mvar|X<sub>n</sub>}} converges '''almost surely''' or '''almost everywhere''' or '''with probability 1''' or '''strongly''' towards ''X'' means that
<math display="block">\mathbb{P}\!\left( \lim_{n\to\infty}\! X_n = X \right) = 1.</math>

This means that the values of {{mvar|X<sub>n</sub>}} approach the value of ''X'', in the sense that events for which {{mvar|X<sub>n</sub>}} does not converge to ''X'' have probability 0 (see ''[[Almost surely]]''). Using the probability space <math>(\Omega, \mathcal{F}, \mathbb{P} )</math> and the concept of the random variable as a function from Ω to '''R''', this is equivalent to the statement
<math display="block">\mathbb{P}\Bigl( \omega \in \Omega: \lim_{n \to \infty} X_n(\omega) = X(\omega) \Bigr) = 1.</math>

Using the notion of the [[Limit superior and limit inferior#Special case: discrete metric|limit superior of a sequence of sets]], almost sure convergence can also be defined as follows:
<math display="block">\mathbb{P}\Bigl( \limsup_{n\to\infty} \bigl\{\omega \in \Omega: | X_n(\omega) - X(\omega) | > \varepsilon \bigr\} \Bigr) = 0 \quad\text{for all}\quad \varepsilon>0.</math>

Almost sure convergence is often denoted by adding the letters ''a.s.'' over an arrow indicating convergence:
{{NumBlk|:|<math>\overset{}{X_n \, \xrightarrow{\mathrm{a.s.}} \, X.}</math>|{{EquationRef|3}}}}

For generic [[random element]]s {''X<sub>n</sub>''} on a [[metric space]] <math>(S,d)</math>, convergence almost surely is defined similarly:
<math display="block">\mathbb{P}\Bigl( \omega\in\Omega\colon\, d\big(X_n(\omega),X(\omega)\big)\,\underset{n\to\infty}{\longrightarrow}\,0 \Bigr) = 1</math>