Math Help - Sum of t distributed variables

Hi, I am looking for a class of fat-tailed probability distributions which would satisfy the following:
Let be two uncorrelated random variables (not necessarily independent). Then it is true that also where


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 .

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

Hey Naena.

You may have to introduce an extra transformation to get what you want. There is an area that is right up your alley called ancillary statistics:

Ancillary statistic - Wikipedia, the free encyclopedia

Take a look at this and see if you can adapt the knowledge and theory of this area to your needs.