Vtyjkyuk

If $X$ follows a Cauchy distribution then $Y = barX = frac1n sum_i=1^n X_i$ 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 $Y=frac 1 nsum_i=1 ^n X_i$, 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.

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

$$
phi_barX_n(t)=[phi_X(t/n)]^n
$$
with $phi_X$ the c.f. of a single r.v. For stable distributions with location parameter zero, we have
$$
phi_X(t)=exp-,
$$
where
$$
Phi=begincasestanleft(fracpialpha2right)&alphaneq1\-frac2pilog|t|&alpha=1endcases
$$
The Cauchy distribution corresponds to $alpha=1$, $beta=0$, so that $phi_barX_n(t)=phi_X(t)$ indeed for any scale parameter $c>0$.

In general,
$$
phi_barX_n(t)=expleftcfractnright,
$$
To get $phi_barX_n(t)=phi_X(t)$, $alpha=1$ seems called for, so
begineqnarray*
phi_barX_n(t)&=&expleftfractnright\
&=&expleftctright,
endeqnarray*
but
$$
logleft|fractnright|neqlogleft|tright|
$$