tail event

Tail eventFernando Sanz Gamiz

Definition.

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

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

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 $(\mathrm{\Omega },ℱ)$)

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