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Math Help - Sum of t distributed variables

  1. #1
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    Sum of t distributed variables

    Hi, I am looking for a class of fat-tailed probability distributions $\mathcal{D}$ which would satisfy the following:
    Let $X,Y\in\mathcal{D}$. be two uncorrelated random variables (not necessarily independent). Then it is true that also $Z\in{\mathcal{D}}$ where
    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 $\beta$.

    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
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  2. #2
    MHF Contributor
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    Re: Sum of t distributed variables

    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.
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