Sum of two generalized Laplace (or Gaussian) variables Is there anything nice I can say about the sum of two independent generalized Laplace variables, with different scales and sizes? i.e. are they distributed same as another generalized Laplace variable with some function of the moments, etc.
Edit: the PDF of generalized {Laplace or Gaussian} is: $f(x) = C\exp\{-|(x-\mu)/a|^b\}$ where 
$a$ is the scale and $b$ is the shape ($C$ is a normalization constant).
Thanks.
 A: I'm not sure there's much nice to say about it: I don't recognize the characteristic of $X + Y$, and can't find a way to manipulate it into the same generalized form. 
Taking the characteristic function of generalized normal random variable $X$ as given in (2.2) of Pogany and Nadarajah, noting a change in parameter names to match those used in the question:
$\phi_X(t) = \frac{e^{it\mu}}{\Gamma(1 / b)}\sum\limits_{m=0}^\infty\frac{\Gamma(2m/b+1/b)}{\Gamma(2m+1)}(-1)^m(at)^{2m}$
(I've changed the formula slightly for ease of differentiation, and replaced a factor of 2 that the paper lost in the algebraic manipulation.) 
Then finding the characteristic of the sum as the product of their respective characteristic functions:
$\phi_{X+Y}(t) = \frac{e^{it\mu_X}}{\Gamma(1 / b_X)}\sum\limits_{m=0}^\infty\frac{\Gamma(2m/b_X+1/b_X)}{\Gamma(2m+1)}(-1)^m(a_Xt)^{2m}\frac{e^{it\mu_Y}}{\Gamma(1 / b_Y)}\sum\limits_{m=0}^\infty\frac{\Gamma(2m/b_Y+1/b_Y)}{\Gamma(2m+1)}(-1)^m(a_Yt)^{2m}$
$=\frac{e^{it(\mu_X + \mu_Y)}}{\Gamma(1/b_X)\Gamma(1/b_Y)}\sum\limits_{m=0}^\infty\frac{\Gamma(2m/b_X+1/b_X)}{\Gamma(2m+1)}(-1)^m(a_Xt)^{2m}\sum\limits_{m=0}^\infty\frac{\Gamma(2m/b_Y+1/b_Y)}{\Gamma(2m+1)}(-1)^m(a_Yt)^{2m}$
We can use that to find the moments of $X+Y$, but as guy points out, that tells us nothing we don't already know from the independence of $X,Y$.
A: I have actually encountered a paper that answers my question, and the answer is, as users have noted, that the result is not GGD. However, one can say something about the similarity of the sum to a GGD.

Qian Zhao; Hong-wei Li; Yuan-tong Shen, "On the sum of generalized
  Gaussian random signals," Signal Processing, 2004. Proceedings. ICSP
  '04. 2004

