# tail event

Tail eventFernando 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.

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

 $\{\lim\sup X_{n}
###### 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 $(\Omega,\mathcal{F})$)

Title tail event TailEvent 2013-03-22 17:07:18 2013-03-22 17:07:18 fernsanz (8869) fernsanz (8869) 9 fernsanz (8869) Definition msc 28A05 SigmaAlgebra KolmogorovZeroOneLaw tail sigma algebra