Kolmogorov zero-one law

Kolomogorov zero-one lawFernando Sanz Gamiz

Theorem (Kolmogorov).

Let $\Omega$ be a set, $\mathcal{F}$ a sigma-algebra of subsets of $\Omega$ and $P$ a probability measure. Given the independent random variables $\{X_{n},n\in\mathbb{N}\}$ , defined on $(\Omega,\mathcal{F},P)$ , it happens that

P(A)=0\mbox{ or }P(A)=1,A\in\mathcal{F}_{\infty},

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

Proof.

Define $\mathcal{F}_{n}=\sigma(X_{1},X_{2},...,X_{n})$ . As any event in $\sigma(X_{n+1},X_{n+2},...)$ is independent of any event in $\sigma(X_{1},X_{2},...,X_{n})$ 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 $\sigma$ -algebra $\mathcal{F}_{\infty}$ is independent of any event in $\bigcup_{n=1}^{\infty}\mathcal{F}_{n}$ ; hence, any event in $\mathcal{F}_{\infty}$ is independent of any event in $\sigma(\bigcup_{n=1}^{\infty}\mathcal{F}_{n})$ ²²again by application of the Monotone Class Theorem. But $\mathcal{F}_{\infty}\subset\sigma(\bigcup_{n=1}^{\infty}\mathcal{F}_{n})$ ³³because $\mathcal{F}_{\infty}\subset\sigma(X_{1},X_{2},...)=\sigma(\bigcup_{n=1}^{% \infty}\mathcal{F}_{n})$ , this last equality being easily proved, so any tail event is independent of itself, i.e., $P(A)=P(A\cap A)=P(A)P(A)$ 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