Editing Convergence of random variables (section)

== Almost sure convergence ==
{{Infobox
  | title = Examples of almost sure convergence
  | bodystyle = width: 28em;
  | headerstyle = background-color: lightblue; text-align:left;
  | datastyle = text-align: left;
  | header1 = Example 1
  | data2 = Consider an animal of some short-lived species. We record the amount of food that this animal consumes per day. This sequence of numbers will be unpredictable, but we may be ''quite certain'' that one day the number will become zero, and will stay zero forever after.
  | header3 = Example 2
  | data4 = Consider a man who tosses seven coins every morning. Each afternoon, he donates one pound to a charity for each head that appeared. The first time the result is all tails, however, he will stop permanently.<br /><br />Let ''X''<sub>1</sub>, ''X''<sub>2</sub>, … be the daily amounts the charity received from him.<br /><br />We may be ''almost sure'' that one day this amount will be zero, and stay zero forever after that.<br /><br />However, when we consider ''any finite number'' of days,  there is a nonzero probability the terminating condition will not occur.
}}

This is the type of stochastic convergence that is most similar to [[pointwise convergence]] known from elementary [[real analysis]].

===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>

===Properties===
* Almost sure convergence implies convergence in probability (by [[Fatou's lemma]]), and hence implies convergence in distribution.  It is the notion of convergence used in the strong [[law of large numbers]].
* The concept of almost sure convergence does not come from a [[topology]] on the space of random variables. This means there is no topology on the space of random variables such that the almost surely convergent sequences are exactly the converging sequences with respect to that topology. In particular, there is no metric of almost sure convergence.

===Counterexamples===
Consider a sequence <math>\{X_n\}</math> of independent random variables such that <math>P(X_n=1)=\frac{1}{n}</math> and <math>P(X_n=0)=1-\frac{1}{n}</math>. For <math>0<\varepsilon<1/2</math> we have <math>P(|X_n|\geq \varepsilon)=\frac{1}{n}</math> which converges to <math>0</math> hence <math>X_n\to 0</math> in probability.

Since <math>\sum_{n\geq 1}P(X_n=1)\to\infty</math> and the events <math>\{X_n=1\}</math> are independent, [[Borel–Cantelli_lemma#Converse-result|second Borel Cantelli Lemma]] ensures that <math>P(\limsup_n \{X_n=1\})=1</math> hence the sequence <math>\{X_n\}</math> does not converge to <math>0</math> almost everywhere (in fact the set on which this sequence does not converge to <math>0</math> has probability <math>1</math>).