One way to generate a Laplace random variable is to generate two IID (independent and identically distributed) exponential random variables and then subtract them:
x_i = y_i - z_i
with y_i and z_i ~ exponential(parameter=b), and of course everything independent.
Then the sum of the x_i is simply (sum y_i) - (sum z_i); each of those two sums have Gamma distributions. To be more specific, since we are summing an integer number of terms, they have Erlang distributions. The difference of two Gammas is called "bilateral gamma", and there are a few papers out there on it. A quick search just found:
Bilateral gamma distributions and processes in financial mathematics
Uwe Küchlera, Stefan Tappe
On the shapes of bilateral Gamma densities
Uwe Küchlera, Stefan Tappe
It would be nice if someone would write a Wikipedia article about bilateral Gammas, I guess.
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