Convergence of Random Series

Theorem 2.5.3 ( Kollmogorov's 0-1 law ) If $X_1,\dots$ are independent and $A\in\mathcal{T}$ then $P(A)=0$ or $1$.

Proof)

(a) $A\in \sigma (X_1,\dots,X_k)$ and $B\in\sigma (X_{k+1},X_{k+2},\dots)$ are independent.

Since $\sigma (X_1,\dots,X_k)$ and $\cup_J \sigma (X_{k+1},\dots,X_{k+j})$ are $\pi$ systems.

(b) $A\in \sigma (X_1,\dots)$ and $B\in \mathcal{T}$ are independent.

$\cup_k \sigma (X_1,\dots,\X_k)$ and $\mathcal{T}$ are $\pi$-systems and therefore independent.

 

Definition 1 ( Permutable ) 

Event $A$ is permutable if $\pi^{-1}A = \{ w : \pi w \in A \}$

 

Theorem 2.5.4 ( Hewitt-Savage 0-1 law ) 

If $X_1,\dots$ are i.i.d and $A\in \mathcal{E}$ then $P(A)\in \{0,1\}$

Proof)

Let $A\in\sigma (X_1,\dots,X_n)$ so that $$P(A_n\Delta A) \rightarrow 0$$

Fix some $\pi$ such that $\pi^2$ is identity and $\pi A_n' = A_n$

Let $A_n'= \{ (X_{n+1},\dots, X_{2n} \in B_n \}$.

Due to i.i.d assumption

$P(A_n' \Delta A)= P(\pi (A_n' \Delta A)) = P(A_n \Delta A)$

Therefore $P(A_n \Delta A) = P(A_n' \Delta A) $

$$P(A_n \Delta A_n' ) \leq P(A_n \Delta A) + P(A\Delta A_n')$$

$ 0 \leq P(A_n) - P( A_n \cap A_n') \leq P(A_n \cup A_n') - P( A_n \cap A_n') \rightarrow 0$

Therefore $P(A_n \cap A_n') \rightarrow P(A)$.