Hi, I am looking for a class of fat-tailed probability distributions $\displaystyle $\mathcal{D}$ $ which would satisfy the following:

Let $\displaystyle $X,Y\in\mathcal{D}$.$ be two uncorrelated random variables (not necessarily independent). Then it is true that also $\displaystyle $Z\in{\mathcal{D}}$$ where

$\displaystyle Z=\sqrt{\beta}X + \sqrt{1-\beta}Y \ \ \ \forall\beta\in[0,1]$

In other words, I am trying to find a fat-tailed distribution that X,Y would be from for which no matter the value of beta, Z would always follow the same distribution. For example normal distribution satisfies this property.

I tried t-distribution. However, for t-distributed X and Y, distribution of Z depends on $\displaystyle $\beta$$.

I was thinking about multivariate t-distribution such that X,Y are not independent but only uncorrelated but I cannot seem to get it right. Any ideas? Links? Papers?

Thanks guys