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Suppose $\{\epsilon_n \vert n \in \mathbb{Z}\}$ is an i.i.d. collection of $L_2$ random variables and $\sum_{i=0}^\infty \vert \varphi_i\vert < \infty $ such that the time series $ \sum_{i=0}^\infty \varphi_i \epsilon_{n-i} $ converges almost surely and in $L_2$ for every $n \in \mathbb{Z}$.

Is it true that $\epsilon_{n+1}$ is independent of $\sum_{i=0}^\infty \varphi_i \epsilon_{n-i} $?

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  • $\begingroup$ Are the $\phi_i$ random variables, too? $\endgroup$ – kimchi lover Oct 9 at 21:29
  • $\begingroup$ No, just real numbers. $\endgroup$ – M. Ost Oct 10 at 12:45
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Since the $\epsilon_k$ are iid, any function of $\{\epsilon_k:k\in A\}$ is independent of any function of $\{\epsilon_k : k\in B\}$, when $A$ and $B$ are disjoint subsets of $\mathbb Z$. That is, the two sigma field generated the two sets of variables are independent. In your case, consider $A=\{n+1\}$ and $B=\{n-i:i\ge 0\}$.

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