From a Laplace distribution $Laplace(0, b)$, we randomly pick one sample from the distribution and repeat $n$ times. Then, we sum these values and get a sum $s = \Sigma_{i=1}^n x_i$ where $_$∼$Laplace(0, b)$.
What's the pdf of this distribution of $f(s)$?
In my understanding, the characteristic distribution of $n$ i.i.d. Laplace variates is $f(s) = \phi(t) = \frac{1}{(1+b^2t^2)^n}$. Based on this characteristic distribution, how can I get the pdf of $f(s)$? Is there a way to write the pdf in terms of Gamma distribution?