Given the following two R.V.s

$$z_{1} = \frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$

and

$$z_{2} = \frac{x_{2}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$

where $x_{i} \sim \mathcal{CN}(0,a), \forall i$ and $a > 0$. As can be seen, the denominator follows a Chi-square distribution with $2M$ degrees of freedom as $x_{i}$ are i.i.d. R.V.s.

Based on these results (1) and (2) and on the observation that for $𝑀>5$ the real and imaginary parts of $z_{i} \forall i$ are normally distributed with mean equal to 0, can we say that $z_{1}$ and $z_{2}$ are independent?

Given the following two R.V.s

$$z_1 = \frac{x_1}{|x_1|^2 + |x_2|^2 + \cdots + |x_M|^2}$$

and

$$z_2 = \frac{x_2}{|x_1|^2 + |x_2|^2 + \cdots + |x_M|^2}$$

where $x_i \sim \mathcal{CN}(0,a), \forall i$ and $a > 0$. As can be seen, the denominator follows a Chi-square distribution with $2M$ degrees of freedom as $x_i$ are i.i.d. R.V.s.

Based on these results (1) and (2) and on the observation that for $𝑀>5$ the real and imaginary parts of $z_i \forall i$ are normally distributed with mean equal to $0,$ can we say that $z_{1}$ and $z_2$ are independent?