PlanetMath: tail event

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tail event

(Definition)

Definition 1 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

$\displaystyle \mathcal F_{\infty}=\bigcap^{\infty}_{n=1}\sigma (X_n,X_{n+1},\cdots)$

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

$\displaystyle \{\lim \sup X_n <c \}, \sum X_n$ converges $\displaystyle , \lim X_n$ exists

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 $(\Omega,\mathcal F)$ )

"tail event" is owned by fernsanz.

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See Also: $\sigma$ -algebra, Kolmogorov zero-one law

Also defines:

tail sigma algebra

Keywords:

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

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Cross-references: probability measure, independent, Kolmogorov zero-one law, theory, sequence, finite, induced, events, measurable space, random variables, subsets, sigma algebra
There is 1 reference to this entry.

This is version 6 of tail event, born on 2007-05-21, modified 2007-05-21.
Object id is 9424, canonical name is TailEvent.
Accessed 1121 times total.

Classification:

AMS MSC:

28A05 (Measure and integration :: Classical measure theory :: Classes of sets , measurable sets, Suslin sets, analytic sets)

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