Where $X$ is a r.v. following a symmetric $T$ distribution with $0 $mean and tail parameter $\alpha$.
I am looking for the distribution of the $n$-summed independent variables $ \sum_{1 \leq i \leq n}|x_i|$.
$Y=|X|$ has for PDF $\frac{2 \left(\frac{\alpha }{\alpha +y^2}\right)^{\frac{\alpha +1}{2}}}{\sqrt{\alpha } B\left(\frac{\alpha }{2},\frac{1}{2}\right)}$, $y \geq 0 $. I managed to get the characteristic function $C(t)$ but could not invert the convolution, that is, $C(t)^n$. Thank you for the help.
