A special class of random variables

Question

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.

Answer 1

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.

Answer 2

Kostya_I is correct, of course, but taking the closure under convolution can sometimes be a bit ugly. If you're interested in a simple, parametric family, consider the hyperbolic secant distribution. I believe if you take the mean-zero sub-family, it satisifies your requirements.