Closed families of random variables exists?

Question

If $\psi$ and $\nu$ are real random variables with normal distribution, their sum is also normal. Do we have such families on a half line or, more importantly on $[a,b]$? (I.e., the value of the random variables should be in [a,b].) We do not want addition to take us out of the family.

Answer 1

On the half line there is always the gamma family: the sum of independent Gamma($a$,1) and Gamma($b$,1) random variables is Gamma($a+b$,1)-distributed.