- what families of distributions can satisfy that the sum of their any two random variables still have a distribution in the same family?
- what families of distributions can satisfy that the product of their any two random variables still have a distribution in the same family?
My questions arose when I read this reply
The sum of normal (Cauchy, Levy) random variables is normal (Cauchy, Levy).
The sum of gamma random variables is gamma if the distributions have a common scale parameter.
The product of log-normal random variables is log-normal.
I know it is not true that the sum of two normal distributed random variables is still normally distributed. For example, if $X$ is normally distributed, define $Y$ to be $X$ if $|X| > c$ and $Y = −X$ if $|X| < c$, where $c > 0$. Then $X+Y$ is not normally distributed.
I am not sure how to verify if the other claims are right or not.
What can be say about a family of distribution that satisfies the above requirements?
Thanks and regards!