Editing Convergence of random variables (section)

== Sure convergence or pointwise convergence ==
To say that the sequence of [[random variables]] (''X''<sub>''n''</sub>) defined over the same [[probability space]] (i.e., a [[random process]]) converges '''surely''' or '''everywhere''' or '''pointwise''' towards ''X'' means

<math display="block">\forall \omega \in \Omega \colon \ \lim_{n\to\infty} X_n(\omega) = X(\omega),</math>

where Ω is the [[sample space]] of the underlying [[probability space]] over which the random variables are defined.

This is the notion of [[pointwise convergence]] of a sequence of functions extended to a sequence of [[random variables]]. (Note that random variables themselves are functions).

<math display="block">\left\{\omega \in \Omega : \lim_{n \to \infty}X_n(\omega) = X(\omega) \right\} = \Omega.</math>

Sure convergence of a random variable implies all the other kinds of convergence stated above, but there is no payoff in [[probability theory]] by using sure convergence compared to using almost sure convergence. The difference between the two only exists on sets with probability zero. This is why the concept of sure convergence of random variables is very rarely used.