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If X and Y follow independently a density distribution represented by the function $\tfrac{1}{\pi} K_0\left(\tfrac{|x|}{a^2}\right)$ (a modified Bessel function of the second kind), then the sum $Z = X + Y$ seems to follow a Laplace double exponential distribution, which for reminder is $L(z) = \tfrac{1}{2a^2} \exp\left(-\tfrac{|z|}{a^2}\right)$.

Now, where I can find any demonstration of that property ?

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  • $\begingroup$ Hello I am studying the same problem, practicing in MGF. In the MGF of the modified Bessel function of second kind I come up with the factor of arccos(t) in the numerator which I cannot see in your answers. Is it something that I am missing?? $\endgroup$
    – Marios
    Commented Feb 12, 2018 at 9:20

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This an easy exercise using moment generating functions. By scaling we can assume WLOG that $a = 1$. Then the MGF of the modified Bessel distribution is $$ E[e^{tX}] = \frac{1}{\sqrt{1-t^2}}, $$ and hence the MGF of the sum of two independent such variables is $\frac{1}{1 - t^2}$, which is the MGF of the Laplace distribution.

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  • $\begingroup$ My godness, I am so stupid... sorry and thanks to refresh my mind. JE $\endgroup$
    – Jean-Eric
    Commented Apr 19, 2013 at 11:17

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