There are quite a lot of study on the normal mixture distributions, say, $X=Y*Z$,where $Z$ is a normal r.v. and Y is a r.v. follows other distributions and $Y$ and $Z$ are independent. Some well-known distributions are Normal inverse Gaussian, generalized Hyperbolic etc.
I would like to know that in probability, is there some other mixture distributions in such form (product of two independent r.v.s) that are well-studied? For example, let $Z$ is possion or exponential and then is there some well-studied probablity distribution can be generated from this form.
If possible, could anyone so kind give me some textbook or monograph on this topic? Thanks so much!!