# criterion for almost-sure convergence

Let ${X}_{1},{X}_{2},\mathrm{\dots}$ and $X$ be random variables^{}.
If, for every $\u03f5>0$, the sum ${\sum}_{n=1}^{\mathrm{\infty}}\mathbb{P}(|{X}_{n}-X|>\u03f5)$ is finite,
then ${X}_{n}$ converge to $X$ almost surely.

###### Proof.

By the Borel-Cantelli lemma^{}, we have $\mathbb{P}({lim\; sup}_{n}\{|{X}_{n}-X|>\u03f5\})=0$.
But ${lim\; sup}_{n}\{|{X}_{n}-X|>\u03f5\}$ is the same as the event $\{{lim\; sup}_{n}|{X}_{n}-X|>\u03f5\}$.
(The latter event involves the limit superior of *numbers* (http://planetmath.org/LimitSuperior); the former involves the
limit superior of *sets* (http://planetmath.org/InfinitelyOften).)
So taking the limit $\u03f5\searrow 0$,
we have $\mathbb{P}({lim\; sup}_{n}|{X}_{n}-X|>0)=0$,
or equivalently
$\mathbb{P}({lim\; sup}_{n}|{X}_{n}-X|=0)=1$.
∎

Title | criterion for almost-sure convergence |
---|---|

Canonical name | CriterionForAlmostsureConvergence |

Date of creation | 2013-03-22 15:54:45 |

Last modified on | 2013-03-22 15:54:45 |

Owner | stevecheng (10074) |

Last modified by | stevecheng (10074) |

Numerical id | 15 |

Author | stevecheng (10074) |

Entry type | Corollary |

Classification | msc 60A99 |

Synonym | corollary of Borel-Cantelli lemma |