# Sum of two different freedom degree multivariate t dsitributions

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).