Kolmogorov zero-one law

Kolomogorov zero-one lawFernando Sanz Gamiz

Theorem (Kolmogorov).

Let Ω be a set, F a sigma-algebra of subsets of Ω and P a probability measureMathworldPlanetmath. Given the independent random variablesMathworldPlanetmath {Xn,nN}, defined on (Ω,F,P), it happens that

P(A)=0 or P(A)=1,A,

i.e.,the probability of any tail event is 0 or 1.


Define n=σ(X1,X2,,Xn). As any event in σ(Xn+1,Xn+2,) is independent of any event in σ(X1,X2,,Xn) 11this assertion should be proved actually, because independence of random variables is defined for every finite number of them and we are dealing with events involving an infinite number. By two successive applications of the Monotone Class Theorem, one can readily prove this is in fact correct, any event in the tail σ-algebra is independent of any event in n=1n; hence, any event in is independent of any event in σ(n=1n) 22again by application of the Monotone Class Theorem. But σ(n=1n) 33because σ(X1,X2,)=σ(n=1n), this last equality being easily proved, so any tail event is independent of itself, i.e., P(A)=P(AA)=P(A)P(A) which implies P(A)=0 or P(A)=1. ∎

Title Kolmogorov zero-one law
Canonical name KolmogorovZerooneLaw
Date of creation 2013-03-22 17:07:21
Last modified on 2013-03-22 17:07:21
Owner fernsanz (8869)
Last modified by fernsanz (8869)
Numerical id 10
Author fernsanz (8869)
Entry type Definition
Classification msc 28A05
Related topic TailEvent