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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 $\frac{2 \left(\frac{\alpha }{\alpha +y^2}\right)^{\frac{\alpha +1}{2}}}{\sqrt{\alpha } B\left(\frac{\alpha }{2},\frac{1}{2}\right)}$, $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.

kjetil b halvorsen
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Nero
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    When $\alpha$ is a positive odd integer, the PDF is a linear combination of the inverse tangent of $x/\sqrt\alpha$ and the logarithm of $\alpha+x^2$ whose coefficients are $\mathbb Z$-rational functions of $x,$ $\sqrt\alpha,$ and $\log\alpha.$ All other values of $\alpha$ look intractable. Thus, one might be content to have the characteristic function for analysis and otherwise perform numerical calculations of the CDF or PDF for practical work. – whuber Aug 24 '18 at 13:21
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    I think this is very relevant https://stats.stackexchange.com/questions/10856/what-is-the-distribution-of-the-difference-of-two-t-distributions because the distribution of |X| is 2 times the distribution of X and over the positive reals instead of the whole real line. – papgeo Aug 24 '18 at 22:09

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