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I want to find out whether there is a concise expression (i.e. not a convolution) for the distribution of a random variable A which is the sum of $n$ i.i.d. rv's $B_i$, which are themselves double exponentially distributed with density $f_B(x) = 1_{x>0} \, p \, \eta^+ e^{-\eta^+ x} + 1_{x<0} \, q \, \eta^- e^{\eta^- x}$ , where $p+q = 1, p \ge 0, q \ge 0, \eta^+ > 0, \eta^- > 0$.

One route of thought went the way from the distribution of a sum of exponential rv's $C_i$ (with density $f_C(x) = 1_{x>0} \, \eta \, e^{-\eta x}, \, \eta > 0$) being a gamma distribution; so - is there anything similar?

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  • $\begingroup$ I assume $\eta^+<0$ is a typo? $\endgroup$ – kjetil b halvorsen Mar 13 '15 at 21:28
  • $\begingroup$ The answer is probably no, if you want an exact result. We could find a good approximation, if that is enough ... $\endgroup$ – kjetil b halvorsen Mar 13 '15 at 22:18
  • $\begingroup$ Yes thanks, typo corrected. $\endgroup$ – Futurist Mar 14 '15 at 8:50
  • $\begingroup$ Related: mathoverflow.net/questions/6887/… $\endgroup$ – kjetil b halvorsen Mar 15 '15 at 0:22
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    $\begingroup$ @kjetil b halvorsen: ok, I couldn't find any closed-form expression; the major hurdle is the characteristic function of the density, even if $p=q=1/2$. I will post a separate question on this. What would be a good approximation? $\endgroup$ – Futurist Apr 20 '15 at 9:02

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