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Let $(X_k)$ be a sequence of independent random variables, all defined on the probability space $(\Omega,\mathcal{A},P)$. Fix a positive integer $n$ and define $$ C_n=\sigma(X_1,X_2,\cdots,X_n),\quad D_n=\sigma(X_{n+1},X_{n+2},\cdots). $$

Here is my question:

How to show by definition that $C_n$ and $D_n$ are independent?


By the definition of independence of $X_k$, for any finite set of positive integers $I$, the $\sigma$-algebras $(\sigma(X_i))_{i\in I}$ are independent. Eventually, one needs to show that

$$ P(A\cap B)=P(A)P(B) $$ for all $A\in C_n$ and $B\in D_n$. How shall I go on?

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    $\begingroup$ It's a lot of work to type out so I'll just leave a comment, but a theorem in Billingsley's book is a slight generalization of this with essentially the same proof. I believe in appears in chapter four. If you're looking for just a hint, use the $\pi - \lambda$ lemma. $\endgroup$ Aug 30 '16 at 21:58
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    $\begingroup$ Indeed, without knowing what you know and do not know, this is next to impossible to answer. $\endgroup$
    – Did
    Aug 30 '16 at 22:41
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The proof of Kolmogrov's zero-one law in Jacod and Protter's Probability Essentials (Theorem 10.6) takes this statement for granted and it is sort of "cheating".

The proof of Theorem 22.3 (the zero-one law) in the third edition of Billingsley's Probability and Measure gives details about how to prove this statement, which eventually boils down to the Dynkin's $\pi$-$\lambda$ Theorem in the book (Theorem 3.2 page 42).

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