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Hometail event

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# tail event

# Tail event

Fernando Sanz Gamiz

###### Definition.

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

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

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

###### Remark 1.

###### 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 $(\Omega,\mathcal{F})$)

Defines:

tail sigma algebra

Keywords:

sigma algebra, zero-one law, sigma algebra induced by random variables

Related:

SigmaAlgebra, KolmogorovZeroOneLaw

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

28A05*no label found*

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