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there are two multivariate t distributions whose freedom degree $\nu$ are different to each other.

$$ \mathbf{x} \sim \mathcal{T}(\nu_x, \mathbf{0}, \Sigma_x) $$

$$ \mathbf{y} \sim \mathcal{T}(\nu_y, \mathbf{0}, \Sigma_y) $$

Then, consider the sum ot them

$$ \mathbf{z} = \mathbf{x} + \mathbf{y} $$

What the distribution of $\mathbf{z}$ will be?

$$ \mathbf{z} \sim \mathcal{T}(\nu_z, \mathbf{0}, \Sigma_z) $$

How can I express $\nu_z$ and $\Sigma_z$?

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Generally speaking and unlike the normal distribution, the sum of $t$ random variables is not a $t$ random variable. In the univariate case, such a sum has the Behrens–Fisher distribution. You can read some more here (requires access).

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    $\begingroup$ Thank you very much. $\endgroup$
    – user331385
    Aug 2 at 0:15

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