When does the sum of two $t$-distributed random variables follow a $t$ distribution? In the scope of a project, I need to find the sum of two independent $t$-distributions.
I know that in the general case, the sum of two $t$-distributed random variables is not $t$-distributed. However, I would like to know what might be the special cases where the sum of two $t$-distributions is a $t$ distribution.
It would also be nice if someone can give me an example of two such $t$-distributions.

Here is a much less formal approach that (mainly) relies on the fact that moments of order $\nu$ or larger do not exist for Student's t-distributions with $\nu$ degrees of freedom.
If $X_\nu$ and $Y_\nu$ have independent Student's t-distributions with $\nu$ degrees of freedom where the moments of order $\nu$ or higher don't exist, then for the sum ($X_\nu +Y_\nu$) will have the same moments that exist and the same moments that don't exist.  That implies that if the sum were to have a Student's t-distribution, then that distribution would also need to have exactly $\nu$ degrees of freedom.
But for $\nu >2$ the variance is $\nu/(\nu-2)$ for a Student's t-distribution and the sum would have variance $2\nu/(\nu-2)$.  But those two variances can't be the same for $\nu>2$.  So for $\nu>2$, there is no sum of two independent Student's t-distributions degrees of freedom $\nu$ that results in a Student's t-distribution.
That leaves only $\nu=1$ and $\nu=2$ to check.
