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I'm wondering if there's a distribution $\Gamma$ such that if we draw $x_i \dots d_n$ iid from $\Gamma$, then $\sum_{i=1}^n |x_i|$ has a nice distribution. I thought maybe the normal distribution, since then the absolute value would be half-normal, but I'm not sure what the distribution of the sum of half-normal variables are.

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If you're willing to consider a distribution that's supported on the positive reals to begin with, there's a nice expression for the distribution of a sum of independent exponential random variables ( – Julian Rosen Sep 20 '12 at 4:18
up vote 2 down vote accepted

If i.i.d. random variables $X_1, X_2, \ldots, X_n$ have Laplacian density $$f(x) = \frac{1}{2}\exp\left(-|x|\right), -\infty < x < \infty,$$ then $|X_i|$ are i.i.d. exponential random variables with mean $1$, and $\sum_{i=1}^n |X_i|$ is a Gamma random variable with mean $n$ and order parameter $n$. This is effectively the same result as the one in the comment by @PinkElephants but without the requirement of support on the positive reals only (which makes the distinction between $X_i$ and $|X_i|$ irrelevant.)

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I think the cases given thus far all satisfy the conditions for the central limit theorem. So for large n there is a normal distribution approximation that should be pretty good and can be normalized so that a standard normal approximation can be used. – Michael Chernick Sep 20 '12 at 19:48

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