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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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new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

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Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier