I have a possibly simple question. Let $\{x_i\}_{i=1}^n$ and $\{u_i\}_{i=1}^n$ be two i.i.d. sequences of random variables. I.e. $x_i$ and $u_i$ are both independent and identically distributed over $i$. Then, consider the random variable, \begin{equation} w_i = (x_i - \overline{x})u_i, \end{equation} where $\overline{x} = (1/n) \sum_{i=1}^n x_i$. Is the sequence $\{w_i\}_{i=1}^n$ then an i.i.d. sequence?
Thanks in advance!