# Correlation between r.v.'s following a distribution that is the ration between complex Gaussian and Chi-square r.v.'s

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 $$a > 1$$. 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.

Is it possible to calculate

$$\mathbb{E} \{ z_{1} z_{2}^{*}\},$$

and show that the variables are correlated or not? Not that $$*$$ is the conjugate.

First here, by the Cauchy--Schwarz inequality, $$E|z_1z^*_2|=E|z_1|\,|z^*_2|\le\sqrt{E|z_1|^2E|z^*_2|^2}=E|z_1|^2<\infty,$$ by Addition in response to the modification of the OP's original question. So, $$Ez_1z^*_2$$ exists and is finite. Therefore and because the joint distribution of the pair $$(-z_1,z^*_2)$$ is the same as that of $$(z_1,z^*_2)$$, we conclude that $$Ez_1z^*_2=E(-z_1)z^*_2=0.$$
Detail sdded in response to the OP's comment: The $$x_i$$'s are iid and, for each $$i$$, the distribution of $$-x_i$$ is the same as that of $$x_i$$. So, the joint distribution of $$(-x_1,x_2,\dots,x_M)$$ is the same as that of $$(x_1,x_2,\dots,x_M)$$. Also, $$|-x_1|=|x_1|$$. So, the random pair $$\begin{multline*} (z_1,z^*_2)=g(x_1,x_2,\dots,x_M):= \\ \Big(\frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + \dots + |x_{M}|^2}, \frac{x_{2}}{|x_{1}|^2 + |x_{2}|^2 + \dots + |x_{M}|^2}\Big) \end{multline*}$$ equals $$\begin{multline*} (-z_1,z^*_2)= \Big(\frac{-x_{1}}{|-x_{1}|^2 + |x_{2}|^2 + \dots + |x_{M}|^2}, \frac{x_{2}}{|x_{1}|^2 + |x_{2}|^2 + \dots + |x_{M}|^2}\Big) \\ =g(-x_1,x_2,\dots,x_M) \end{multline*}$$ in distribution, as was stated above.