Let $Z=XY$ where $X$, $Y$ are random variables with support of non-trivial measure. For what distributions of $X$ and $Y$ can $Z$ be guaranteed to be Gaussian?



Which distributions of $X$ and $Y$ yield a Gaussian $Z=XY$?
It was shown in [this paper (see formula (2) there)](http://arxiv.org/abs/1803.09838v1) that any normal random variable (r.v.) $Z$ is multiplicatively infinitely divisible; that is, for each natural $k$ there exist iid r.v.'s $W\_1,\dots,W\_k$ such that $Z$ equals $W\_1\cdots W\_k$ in distribution; the distribution of $W\_1$ is explicitly described. 


From that description, it is easy to get an entire continuous one-parameter family $\big((X\_t,Y\_t)\big)\_{t\in(0,1)}$ of pairs $(X\_t,Y\_t)$ of nontrivial independent r.v.'s such that $X\_tY\_t$ has the (say) standard normal distribution for each $t\in(0,1)$. 


