Ideally, what I'm looking for are two random variables, $ X $ Y $ Y $ (If one is positive, then that is even better) such that $ Z = X cdot Y sim mathcal {N} (0,1) $ where $ X, Y $ There are some distributions from which I can generate samples. If someone knows of a result like that, it would be very helpful.

I believe that existence can be tested implicitly: Suppose $ X, Y $ They have the same distribution. Leave $ m_k $ Be the sequence of Gaussian moments and define. $ tilde {m} _k = sqrt {m_k} $, so it is enough to show that there is a distribution with $ tilde {m} _k $ moments This can be done by showing that the moment sequence converges, so that the moment generation function is well defined and obtains the density of the Laplace transform. Is that correct?