#StackBounty: #probability #distributions #convolution Sum of absolute values of T random variables

Bounty: 50

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 variable $ sum_{1 leq i leq n}|x_i|$.

$Y=|X|$ has for PDF $frac{2 left(frac{alpha }{alpha +y^2}right)^{frac{alpha +1}{2}}}{sqrt{alpha } Bleft(frac{alpha }{2},frac{1}{2}right)}$, $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.

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