# tail event

Tail eventFernando Sanz Gamiz

###### Definition.

Let $\mathrm{\Omega}$ be a set and $\mathcal{F}$ a sigma algebra of subsets
of $\mathrm{\Omega}$. Given the random variables^{} $\{{X}_{n},n\in \mathbb{N}\}$, defined
on the measurable space^{} $(\mathrm{\Omega},\mathcal{F})$, the *tail
events* are the events of the *tail $\sigma $-algebra*

$${\mathcal{F}}_{\mathrm{\infty}}=\bigcap _{n=1}^{\mathrm{\infty}}\sigma ({X}_{n},{X}_{n+1},\mathrm{\cdots})$$ |

where $\sigma ({X}_{n},{X}_{n+1},\mathrm{\cdots})$ is the $\sigma $-algebra induced by $({X}_{n},{X}_{n+1},\mathrm{\cdots})$.

###### Remark 1.

One can intuitively think of tail events as those events whose
ocurrence or not is not affected by altering any finite number of
random variables in the sequence^{}. Some examples are

$$ |

###### Remark 2.

One of the most important theorems in probability theory due to
Kolomogorv, is the Kolmogorov zero-one law which states that, in the case of independent random variables, the
probability of any tail event is 0 or 1 (provided there is a
probability measure^{} defined on $(\mathrm{\Omega},\mathcal{F})$)

Title | tail event |
---|---|

Canonical name | TailEvent |

Date of creation | 2013-03-22 17:07:18 |

Last modified on | 2013-03-22 17:07:18 |

Owner | fernsanz (8869) |

Last modified by | fernsanz (8869) |

Numerical id | 9 |

Author | fernsanz (8869) |

Entry type | Definition |

Classification | msc 28A05 |

Related topic | SigmaAlgebra |

Related topic | KolmogorovZeroOneLaw |

Defines | tail sigma algebra |