Can we characterize random variables $X$ that satisfy
$$
X\sim Y - Z
$$
for two independent **positive** random variables $Y$ and $Z$?

Are $Y$ and $Z$ unique in some sense?

Can (one possible choice of) $Y$ and $Z$ be constructed (e.g. formulas for probability density or characteristic function, or sampling algorithms) when they exist?

*Possibly simpler question:*
Which random variables $X$ satisfy
$$
X\sim Y_1-Y_2
$$
for i.i.d. positive random variables $Y_1\sim Y_2\sim Y$? Since the characteristic function satisfies
$$
\phi_X = \phi_{Y}\overline{\phi_{Y}}
$$
we must have $\phi_{X}\geq 0$ -- is that sufficient? Is $Y$ unique in some sense? Can it be constructed?

For example, Laplace random variables satisfy $\phi_{X} = (1+x^2)^{-1}=(1+ix)^{-1}\overline{(1+ix)^{-1}}=\phi_{Y}\overline{\phi_{Y}}$ where $Y$ is exponential. Exponentials are positive of course, so we got lucky with this particular decomposition and can write $X=Y_1-Y_2$ as desired. Had we picked $(1+x^2)^{-1}=(1+x^2)^{-1/2}\overline{(1+x^2)^{-1/2}}$ this wouldn't have worked out.

This approach slightly generalizes to $\phi_{X}$ that are rational in $x^2$, but not at all (at least not obviously to me) to only slightly different examples like Linnik random variables, where $\phi_{X} = (1+|x|^{\alpha})^{-1}$, or to limits such as normal random variables, where $\phi_{X}=e^{-\sigma^2x^2}$.

The only result I found that goes remotely in this direction was a theorem by Boas and Kac that positive definite functions with compact support have a convolution square root with half-length compact support. This has a support flavor, but a different one than I'm looking for.