A symmetric iid process Let $X_1, X_2, \ldots$ be an iid process with $X_i$ having a symmetric distribution around $0$. Then can I always write
$$X_1 - \alpha X_{t-1}-\alpha^2 X_{t-2}-\cdots \stackrel{iid}{=} X_1 + |\alpha| X_{t-1}+|\alpha|^2 X_{t-2}+\cdots?$$ 
I think so, as $X_i\stackrel{iid}{=}-X_i$ and, thus, I just have to consider the magnitude of $\alpha$, the sign of it will be taken care of by $X_i$'s.
 A: We can use the following facts

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*If $\left(X_i\right)_{i\geqslant 1}$ is an i.i.d. sequence of symmetric around zero random variables and for all $i\geqslant 1$, $a_i$ is a (deterministic) element of $\{-1,1\}$, then the sequences $\left(X_i\right)_{i\geqslant 1}$ and $\left(a_iX_i\right)_{i\geqslant 1}$ have the same distribution.
To see this, it suffices to prove that for all $n$, the vectors $\left(X_i\right)_{i= 1}^n$ and $\left(a_iX_i\right)_{i= 1}^n$ have the same distribution, which can be done by considering the characteristic functions. These ones can be written as a product (by independence), and the factors are equal, by the symmetry around zero.


*If two sequences $\left(X_i\right)_{i\geqslant 1}$ and $\left(Y_i\right)_{i\geqslant 1}$ have the same distribution and $f\colon \mathbb R^{\mathbb N}\to\mathbb R$ is a measurable function (where $\mathbb R^N$ is endowed with the product $\sigma$-algebra, then $f\left( \left(X_i\right)_{i\geqslant 1}   \right)$ and $f\left( \left(Y_i\right)_{i\geqslant 1}   \right)$ have the same distribution.
