I am trying to prove the statement:
If $X\sim\mathcal{N}(0,\sigma_1^2)$ and $Y\sim\mathcal{N}(0,\sigma_2^2)$ are independent random variables,
then $\frac{XY}{\sqrt{X^2+Y^2}}$ is also a Normal random variable.
For the special case $\sigma_1=\sigma_2=\sigma$ (say), we have the well-known result that $\frac{XY}{\sqrt{X^2+Y^2}}\sim\mathcal{N}\left(0,\frac{\sigma^2}{4}\right)$ whenever $X$ and $Y$ are independent $\mathcal{N}(0,\sigma^2)$ variables. In fact, it is more generally known that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}}$ are independent $\mathcal{N}\left(0,\frac{\sigma^2}{4}\right)$ variables.
A proof of the last result follows by using the transformation $(X,Y)\to(R,\Theta)\to(U,V)$ where $x=r\cos\theta,y=r\sin\theta$ and $u=\frac{r}{2}\sin(2\theta),v=\frac{r}{2}\cos(2\theta)$. Indeed, here $U=\frac{XY}{\sqrt{X^2+Y^2}}$ and $V=\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}}$. I tried to imitate this proof for the problem at hand but it appears to get messy.
If I haven't made any error, then for $(u,v)\in\mathbb{R}^2$ I end up with the joint density of $(U,V)$ as
$$f_{U,V}(u,v)=\frac{2}{\sigma_1\sigma_2\pi}\exp\left[-\sqrt{u^2+v^2}\left(\frac{\sqrt{u^2+v^2}+v}{\sigma_1^2}+\frac{\sqrt{u^2+v^2}-v}{\sigma_2^2}\right)\right]$$
I have the multiplier $2$ above as the transformation is not one-to-one.
So density of $U$ would be given by $\displaystyle \int_{\mathbb{R}}f_{U,V}(u,v)\,\mathrm{d}v$, which isn't readily evaluated.
Now I am interested to know if there is a proof where I can only work with $U$ and don't have to consider some $V$ to show that $U$ is Normal. Finding the CDF of $U$ doesn't look so promising to me at the moment. I would also like to do the same for the case $\sigma_1=\sigma_2=\sigma$.
That is, if $X$ and $Y$ are independent $\mathcal{N}(0,\sigma^2)$ variables then I wish to show that $Z=\frac{2XY}{\sqrt{X^2+Y^2}}\sim\mathcal{N}(0,\sigma^2)$ without using a change of variables. If somehow I can argue that $Z\stackrel{d}{=}X$, then I'm done. So two questions here, the general case and then the particular case.
Related posts on Math.S.E:
$X^2-Y^2/ \sqrt{X^2+Y^2}\sim N(0,1)$ when $X,Y\sim N(0,1)$ independently.
Edit.
This problem is in fact due to L. Shepp as I found out in the exercises of An Introduction to Probability Theory and Its Applications (Vol. II) by Feller, alongwith a possible hint:
Surely, $U=\frac{XY}{\sqrt{X^2+Y^2}}=\frac{1}{\sqrt{\frac{1}{X^2}+\frac{1}{Y^2}}}$ and I have the density of $\frac{1}{X^2}$ at hand.
Let's see what I could do now. Apart from this, a little help with the integral above is also welcome.
Best Answer
The original solution of the problem by Shepp uses the concept of stable law property, which seems a bit advanced for me at the moment. So I could not comprehend the hint given in the exercise I cited in my post. I guess a proof involving only the single variable $U=\frac{XY}{\sqrt{X^2+Y^2}}$ and not using a change of variables is difficult to come up with. So I share three open access papers I found that provide an alternate solution to the problem:
A note on Normal functions of Normal random variables
Normal functions of Normal random variables
A Result of Shepp
The first one has convinced me not to go down the integration path I took with that choice of the variable $V$ to derive the density of $U$. It is the third paper that looks like something I can follow. I give a brief sketch of the proof here:
We assume without loss of generality $\sigma_1^2=1$, and set $\sigma_2^2=\sigma^2$. Now noting that $X^2\sim\chi^2_1$ and $\frac{Y^2}{\sigma^2}\sim\chi^2_1$ are independent, we have the joint density of $(X^2,Y^2)$. We denote it by $f_{X^2,Y^2}$.
Consider the transformation $(X^2,Y^2)\to(W,Z)$ such that $W=\frac{X^2Y^2}{X^2+Y^2}$ and $Z=\frac{X^2+Y^2}{Y^2}$. So we have the joint density of $(W,Z)$. Let us denote it by $f_{W,Z}$. Following the standard procedure, we integrate $f_{W,Z}$ wrt to $z$ to get the marginal density $f_W$ of $W$.
We find that $W=U^2$ is a Gamma variate with parameters $\frac{1}{2}$ and $2(1+\frac{1}{\sigma})^{-2}$, so that $(1+\frac{1}{\sigma})^2\,W\sim\chi^2_1$. We note that the density of $U$ is symmetric about $0$. This implies that $(1+\frac{1}{\sigma})U\sim\mathcal{N}(0,1)$, and hence $U\sim\mathcal{N}\left(0,\left(\frac{\sigma}{\sigma+1}\right)^2\right)$.