How can I prove that σ($X_{n+1}$) and σ$(X_1,…,X_n)$ are independent, with $X_n$ sequence of independent random variables?
I'm trying to prove this statement using the indipendence of two σ-algebras:
σ($X_{n+1}$) and σ($\mathcal I$)
with $\mathcal I$={$\bigcap^n_{i=0}A_i$, with $A_i\inσ(X_i)$}, using that $\mathcal I$ is a $\pi$-system such that σ($\mathcal I$)=σ($X_1,...,X_n$) but I don't know how to prove this last equality.
What you need is the $\pi -\lambda$ theorem: let $\Lambda_0$ be a $\pi$ system (in the sense that intersection of any two sets in this class is also in this class) and let $\Lambda$ be a $\lambda$ system ( in the sense $A\setminus B$ is in it whenever $A$ and $B$ are in it, and $\cup A_n$ is in it whenever each $A_n$ is in it and $A_n$'s are increasing); if $\Lambda_0 \subset \Lambda$ then $\sigma(\Lambda_0) \subset \Lambda$. [ Ref. Billingsley's Probability and Measure]. To prove the independence of $\sigma(X_{n+1})$ and $\sigma(X_1,X_2,...X_n)$ let $\mathcal C$ be sets of the type $X_{n+1}^{-1}(B)$ where $B$ is a Borel set in $\mathbb R$. Note that $\mathcal C$ is precisely $\sigma (X_{n+1})$. Let $\Lambda_0$ be the collection of all sets of the type $X_1^{-1}(A_1) \cap X_2^{-1}(A_2)\cap ...\cap X_n^{-1}(A_n)$ where each $A_i$ is a Borel set in $\mathbb R$. From independence of $X_1,X_2,...,X_{n+1}$ we see that $P(C \cap D)=P(C)P(D)$ whenever $C \in \mathcal C$ and $D \in \Lambda_0$. Now let $\Lambda$ consist of all sets $E$ for which $P(C \cap E)=P(C)P(E)$ where $C \in \mathcal C$ is fixed. Then easy verifications show that $\Lambda$ is a $\lambda$ system which contains the $\pi$ system $\Lambda_0$. Hence $\sigma(\mathcal \Lambda_0) \subset \Lambda$. To finish the proof you just have to observe that $\sigma(\mathcal \Lambda_0)$ is precisely $\sigma(X_1,X_2,...X_n)$.