tail event

Tail eventFernando Sanz Gamiz


Let Ω be a set and a sigma algebra of subsets of Ω. Given the random variablesMathworldPlanetmath {Xn,n}, defined on the measurable spaceMathworldPlanetmathPlanetmath (Ω,), the tail events are the events of the tail σ-algebra


where σ(Xn,Xn+1,) is the σ-algebra induced by (Xn,Xn+1,).

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 sequenceMathworldPlanetmath. Some examples are

{limsupXn<c},Xnconverges ,limXn 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 measureMathworldPlanetmath defined on (Ω,))

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