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I am trying to formalise the result that the sum of two independent Levy-distributed (having parameter $c$) random variables has also Levy distribution with parameter $4c$.

Idea of the proof By Levy-Hitchin theorem one can derive the characteristic function of Levy distribution and then apply the inverse Fourier transform.

My question is, is there a more intuitive a less computational way to deduce the Levy distribution?

Levy distribution:

$\sqrt{\frac{c}{2\pi}}\frac{e^{-\frac{c}{2x}}}{x^{3/2}}$

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