Are there any distributions other than Cauchy for which the arithmetic mean of a sample follows the same distribution? If $X$ follows a Cauchy distribution then $Y = \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$ also follows exactly the same distribution as $X$; see this thread.


*

*Does this property have a name?

*Are there any other distributions for which this is true?
EDIT
Another way of asking this question:
let $X$ be a random variable with probability density $f(x)$.
let $Y=\frac 1 n\sum_{i=1} ^n X_i$, where $X_i$ denotes the ith observation of $X$.
$Y$ itself can be considered as a random variable, without conditioning on any specific values of $X$.
If $X$ follows a Cauchy distribution, then the probability density function of $Y$ is $f(x)$
Are there any other kinds of (non trivial*) probability density functions for $f(x)$ that result in $Y$ having a probability density function of $f(x)$?
*The only trivial example I can think of is a Dirac delta. i.e. not a random variable.
 A: This is not really an answer, but at least it does not seem to be easy to create such an example from a stable distribution. We would need to produce a r.v. whose characteristic function is the same as that of its average.
In general, for an iid draw, the c.f. of the average is
$$
\phi_{\bar{X}_n}(t)=[\phi_X(t/n)]^n
$$
with $\phi_X$ the c.f. of a single r.v. For stable distributions with location parameter zero, we have
$$
\phi_X(t)=\exp\{-|ct|^\alpha(1-i\beta \text{sgn}(t)\Phi)\},
$$
where
$$
\Phi=\begin{cases}\tan\left(\frac{\pi\alpha}{2}\right)&\alpha\neq1\\-\frac{2}{\pi}\log|t|&\alpha=1\end{cases}
$$
The Cauchy distribution corresponds to $\alpha=1$, $\beta=0$, so that $\phi_{\bar{X}_n}(t)=\phi_X(t)$ indeed for any scale parameter $c>0$. 
In general,
$$
\phi_{\bar{X}_n}(t)=\exp\left\{-n\left|c\frac{t}{n}\right|^\alpha\left(1-i\beta \text{sgn}\left(\frac{t}{n}\right)\Phi\right)\right\},
$$
To get $\phi_{\bar{X}_n}(t)=\phi_X(t)$, $\alpha=1$ seems called for, so 
\begin{eqnarray*}
\phi_{\bar{X}_n}(t)&=&\exp\left\{-n\left|c\frac{t}{n}\right|\left(1-i\beta \text{sgn}\left(\frac{t}{n}\right)\left(-\frac{2}{\pi}\log\left|\frac{t}{n}\right|\right)\right)\right\}\\
&=&\exp\left\{-\left|ct\right|\left(1-i\beta \text{sgn}\left(t\right)\left(-\frac{2}{\pi}\log\left|\frac{t}{n}\right|\right)\right)\right\},
\end{eqnarray*}
but
$$
\log\left|\frac{t}{n}\right|\neq\log\left|t\right|
$$
A: Normal distribution and shifted Poisson are examples. The shifted Poisson is $s=x-\lambda$, where $\lambda$ is Poisson intensity. There's a whole family of distribution such that the linear combination (not just the sample mean) of variables follows the same distribution, it's called stable distribution.
