I have two stochastic processes $\{X_i\}_{i \in \mathbb{Z}}$ and $\{Y_i\}_{i \in \mathbb{Z}}$ which are each of them individually independent and identically distributed. Also they are independent of each other.
I build $Z_n = \theta X_n + (1- \theta) Y_n$ and $W_n = \theta_n X_n + (1- \theta_n) Y_n$ where $\theta$ is the result of a fair coin flip done once and $\theta_n$ is the result of a fair coin flip done at each time $n$.
How can I proof that $\{W_i\}_{i \in \mathbb{Z}}$ is an i.i.d. process whereas $\{Z_i\}_{i \in \mathbb{Z}}$ is not?