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 $\Sigma_z$?


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).

  • 1
    $\begingroup$ Thank you very much. $\endgroup$
    – user331385
    Aug 2 at 0:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.