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 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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