# Independence of a convergent series of i.i.d. random variables

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 $$\sum_{i=0}^\infty \varphi_i \epsilon_{n-i}$$?

• Are the $\phi_i$ random variables, too? – kimchi lover Oct 9 at 21:29
• No, just real numbers. – M. Ost Oct 10 at 12:45

## 1 Answer

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 $$B=\{n-i:i\ge 0\}$$.