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 :)
Re: Sum of t distributed variables
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.