Return to Answer

You can use the rotational invariance of the Gaussian distribution to take $\mathbf{y}=2^{-1/2}\sigma_y(\sqrt \xi_{2M},0,0,\ldots,0)$, with $\xi_{2M}$ distributed independently of $\mathbf{x}$ according to a chi-squared distribution with $2M$ degrees of freedom. Then $$Z=\tfrac{1}{2}\sigma_y^2 \xi_{2M}|z|^2\;\;\text{with}\;\; z=\frac{x_1}{\sum_{n=1}^M |x_n|^2}.$$ This can be further reduced to $$Z=\frac{\sigma_y^2}{\sigma_x^2} \frac{\xi_{2M}\xi_{2}}{(\xi_{2}+\xi_{2M-2})^2},$$ with $\xi_2$ and $\xi_{2M-2}$ independently chi-squared distributed with $2$, respectively, $2M-2$ degrees of freedom.

I don't think the distribution of this rational function of three independent chi-squared distributions has a closed form expression. For $M\gg 1$ one has simply $$Z\rightarrow \frac{\sigma_y^2}{2M\sigma_x^2}\xi_2,$$ so $Z$ has for large $M$ a chi-squared distribution with 2 degrees of freedom.

Moments for any $M$:

Upon integration of $Z$ over the independent chi-squared variables $\xi_2$, $\xi_{2M}$, and $\xi_{2M-2}$ I find the expectation value $$\mathbb{E}(Z)=\frac{1}{M-1}(\sigma_y/\sigma_x)^2.$$

You can use the rotational invariance of the Gaussian distribution to take $\mathbf{y}=2^{-1/2}\sigma_y(\sqrt \xi_{2M},0,0,\ldots,0)$, with $\xi_{2M}$ distributed independently of $\mathbf{x}$ according to a chi-squared distribution with $2M$ degrees of freedom. Then $$Z=\tfrac{1}{2}\sigma_y^2 \xi_{2M}|z|^2\;\;\text{with}\;\; z=\frac{x_1}{\sum_{n=1}^M |x_n|^2}.$$ This can be further reduced to $$Z=\frac{\sigma_y^2}{\sigma_x^2} \frac{\xi_{2M}\xi_{2}}{(\xi_{2}+\xi_{2M-2})^2},$$ with $\xi_2$ and $\xi_{2M-2}$ independently chi-squared distributed with $2$, respectively, $2M-2$ degrees of freedom.

I don't think the distribution of this rational function of three independent chi-squared distributions has a closed form expression. For $M\gg 1$ one has simply $$Z\rightarrow \frac{\sigma_y^2}{2M\sigma_x^2}\xi_2,$$ so $Z$ has for large $M$ a chi-squared distribution with 2 degrees of freedom.

You can use the rotational invariance of the Gaussian distribution to take $\mathbf{y}=2^{-1/2}\sigma_y(\sqrt \xi_{2M},0,0,\ldots,0)$, with $\xi_{2M}$ distributed independently of $\mathbf{x}$ according to a chi-squared distribution with $2M$ degrees of freedom. Then $$Z=\tfrac{1}{2}\sigma_y^2 \xi_{2M}|z|^2\;\;\text{with}\;\; z=\frac{x_1}{\sum_{n=1}^M |x_n|^2}.$$ This can be further reduced to $$Z=\frac{\sigma_y^2}{\sigma_x^2} \frac{\xi_{2M}\xi_{2}}{(\xi_{2}+\xi_{2M-2})^2},$$ with $\xi_2$ and $\xi_{2M-2}$ independently chi-squared distributed with $2$, respectively, $2M-2$ degrees of freedom.

I don't think the distribution of this rational function of three independent chi-squared distributions has a closed form expression. For $M\gg 1$ one has simply $$Z\rightarrow \frac{\sigma_y^2}{2M\sigma_x^2}\xi_2,$$ so $Z$ has for large $M$ a chi-squared distribution with 2 degrees of freedom.

Moments for any $M$:

Upon integration of $Z$ over $\xi_2$, $\xi_{2M}$, and $\xi_{2M-2}$ I find the expectation value $$\mathbb{E}(Z)=\frac{1}{M-1}(\sigma_y/\sigma_x)^2.$$

You can use the rotational invariance of the Gaussian distribution to take $\mathbf{y}=2^{-1/2}\sigma_y(\sqrt \xi_{2M},0,0,\ldots,0)$, with $\xi_{2M}$ distributed independently of $\mathbf{x}$ according to a chi-squared distribution with $2M$ degrees of freedom. Then $$Z=\tfrac{1}{2}\sigma_y^2 \xi_{2M}|z|^2\;\;\text{with}\;\; z=\frac{x_1}{\sum_{n=1}^M |x_n|^2}.$$ This can be further reduced to $$Z=\frac{\sigma_y^2}{\sigma_x^2} \frac{\xi_{2M}\xi_{2}}{(\xi_{2}+\xi_{2M-2})^2},$$ with $\xi_2$ and $\xi_{2M-2}$ independently chi-squared distributed with $2$, respectively, $2M-2$ degrees of freedom.

I don't think the distribution of this rational function of three independent chi-squared distributions has a closed form expression. For $M\gg 1$ one has simply $$Z\rightarrow \frac{\sigma_y^2}{2M\sigma_x^2}\xi_2,$$ so $Z$ has for large $M$ a chi-squared distribution with 2 degrees of freedom.

Moments for any $M$:

Upon integration of $Z$ over the independent chi-squared variables $\xi_2$, $\xi_{2M}$, and $\xi_{2M-2}$ I find the expectation value $$\mathbb{E}(Z)=\frac{1}{M-1}(\sigma_y/\sigma_x)^2.$$

You can use the rotational invariance of the Gaussian distribution to take $\mathbf{y}=2^{-1/2}\sigma_y(\sqrt \xi_{2M},0,0,\ldots,0)$, with $\xi_{2M}$ distributed independently of $\mathbf{x}$ according to a chi-squared distribution with $2M$ degrees of freedom. Then $$Z=\tfrac{1}{2}\sigma_y^2 \xi_{2M}|z|^2\;\;\text{with}\;\; z=\frac{x_1}{\sum_{n=1}^M |x_n|^2}.$$ This can be further reduced to $$Z=\frac{\sigma_y^2}{\sigma_x^2} \frac{\xi_{2M}\xi_{2}}{(\xi_{2}+\xi_{2M-2})^2},$$ with $\xi_2$ and $\xi_{2M-2}$ independently chi-squared distributed with $2$, respectively, $2M-2$ degrees of freedom.

I don't think the distribution of this rational function of three independent chi-squared distributions has a closed form expression. For $M\gg 1$ one has simply $$Z\rightarrow \frac{\sigma_y^2}{2M\sigma_x^2}\xi_2,$$ so $Z$ has a chi-squared distribution with 2 degrees of freedom.

You can use the rotational invariance of the Gaussian distribution to take $\mathbf{y}=2^{-1/2}\sigma_y(\sqrt \xi_{2M},0,0,\ldots,0)$, with $\xi_{2M}$ distributed independently of $\mathbf{x}$ according to a chi-squared distribution with $2M$ degrees of freedom. Then $$Z=\tfrac{1}{2}\sigma_y^2 \xi_{2M}|z|^2\;\;\text{with}\;\; z=\frac{x_1}{\sum_{n=1}^M |x_n|^2}.$$ This can be further reduced to $$Z=\frac{\sigma_y^2}{\sigma_x^2} \frac{\xi_{2M}\xi_{2}}{(\xi_{2}+\xi_{2M-2})^2},$$ with $\xi_2$ and $\xi_{2M-2}$ independently chi-squared distributed with $2$, respectively, $2M-2$ degrees of freedom.

I don't think the distribution of this rational function of three independent chi-squared distributions has a closed form expression. For $M\gg 1$ one has simply $$Z\rightarrow \frac{\sigma_y^2}{2M\sigma_x^2}\xi_2,$$ so $Z$ has for large $M$ a chi-squared distribution with 2 degrees of freedom.

Moments for any $M$:

Upon integration of $Z$ over $\xi_2$, $\xi_{2M}$, and $\xi_{2M-2}$ I find the expectation value $$\mathbb{E}(Z)=\frac{1}{M-1}(\sigma_y/\sigma_x)^2.$$