# Formula for the nth convolution of a laplace random variable

Let x_1, x_2, ... be iid draws from a laplace distribution with scale parameter b. Is there a relatively nice closed form for x_1+x_2+...x_n? I've seen a derivation floating around for when b=1, but I couldn't figure out a generalisation.

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Did any of you work out the exact expression in terms of the scale parameter b? I'm having trouble taking the inverse Fourier transform :( Thank you! –  user3150 Jan 11 '10 at 7:30

The distribution of the n-th convolution of the Laplace distribution can be computed from the characteristic function: exp(i mu t)/(1+b^2 t^2). (See the Wikipedia article). The characteristic function of the n-th convolution becomes exp(i n mu t)/(1+b^2 t^2)n = exp(i n mu t)/((1 - i b t)^n ((1 + i b t)^n) . The inverse Fourier transform can be computed using the residue theorem. The integration contour is closed from the upper or lower half plane according to the sign of (x-n mu).

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