I'm interested in classes C of $R^1$-valued random variables which possess the following properties:


1) the sum of two independent random variables from class C belongs to class C;


2) for any $\lambda \in R^1$, $\xi \in C$ we have $\lambda\xi \in C$; 


3) any random variable from class $C$ has tails which are heavier than the Gaussian tails;


4) for any $\xi \in C$ we have $\mathrm{E}\xi=0, \mathrm{E}\xi^2<\infty$. 


If we omit the restriction $\mathrm{E}\xi=0$ and restriction No. 2 then there is an obvious example: 
a class of gamma-distributed random variables with one parameter of the distribution fixed. Can anyone suggest other examples? 


Thanks in advance. 


Best wishes, Ievgen. 



A special class of random variables
There are many such classes. For example, take a single symmetric random variable satisfying 3 and 4, and let $C$ consist of all possible variables that you get from it by applying scalings and convolutions. Or let $C$ be the set of of all symmetric random variables for which there are $c\_1,c\_2>0$ and $k\_1,k\_2>3$ such that its density satisfies $$\frac{c\_1}{(1+|x|)^{k\_1}}\leq \rho(x)\leq\frac{c\_2}{(1+|x|)^{k\_2}}$$
Or take all $X-\mathbb{E}X$, where $X$ is exponentially distributed.


