Let $\mathbf{w}$, $\mathbf{e}$ and $\mathbf{f}$ be three independent complex vectors. Suppose that $\mathbf{f}$ and $\mathbf{w}$ are of unit norm. In addition, we assume that $\mathbf{e}$ is complex gaussian with zero mean and variance $= \sigma^2$.
Now, we define : $x=(\mathbf{w}^* \mathbf{f})^*$ and $y=\mathbf{e}^* \mathbf{f}$, where $(\cdot)^*$ denotes the conjugate transpose.
Can we claim that $x$ and $y$ are independent (complex) random variables? if yes, why?