Kolmogorov 0-1 law §

Def Given a collection of random variables $(X_n{)}_{n\ge 1}$ we define the $\sigma$-algebras ${\mathcal{T}}_n$ by ${\mathcal{T}}_n:=\sigma (X_n,X_{n+1},\dots )$. Then the tail $\sigma$-algebra is defined by

$$\mathcal{T}:=\bigcap _{n\ge 1}{\mathcal{T}}_n.$$

Intuition: The events which are in the tail sigma algebra are those which are unaffected by finitely many variables.

Let’s see some examples of such events.

Examples

• $\{{\sum }_{n=1}^{\infty }{X}_n<\infty \}\in \mathcal{T}$ since this is equivalent to having a finite tail:
• $\{{\lim }_{n\to \infty }{X}_n\text{ exists }\}$

Theorem (Kolmogorov) Let $(X_n{)}_{n\ge 1}$ denote a collection of indipendent random variables. Then the tail $\sigma$-algebra is trivial, meaning $\mathbb{P}(A)\in \{0,1\}$ for all $A\in \mathcal{T}$.

Proof Let’s show that $\mathcal{T}$ is indipendent of itself. We introduce the $\sigma$-algebras

$${\mathcal{X}}_n:=\sigma (X_1,X_2,\dots ,X_n)$$ $${\mathcal{T}}_n:=\sigma (X_{n+1},X_{n+2},\dots ).$$

Since the variables are indipendnet, ${\mathcal{X}}_n$ and ${\mathcal{T}}_n$ are indipendent. Since $\mathcal{T}\subseteq {\mathcal{T}}_n$, $\mathcal{T}$ is indipendent of ${\mathcal{X}}_n$ for all $n$. Now let ${\mathcal{X}}_{\infty }:=\sigma ((X_n{)}_{n\ge 1})$. It suffices to show that $\mathcal{T}$ is indipendent of ${\mathcal{X}}_{\infty }$, from the inclusion $\mathcal{T}\subseteq {\mathcal{X}}_{\infty }$ the thesis will follow. To show this indipendence, observe that

$${\mathcal{X}}_{\infty }=\sigma \left(\bigcup _{n\ge 1}{\mathcal{X}}_n\right)$$

and that ${\bigcup }_{n\ge 1}{\mathcal{X}}_n$ is a Pi-system, and $\mathcal{T}$ is clearly indipendent since it is for all ${\mathcal{X}}_n$. $\square$