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It is well known that if $X$ and $Y$ follow i.i.d. Cauchy distribution of scale $\gamma$, say $$ p_{\gamma} (x) = \frac{1}{ \pi \gamma ( 1 + x^2 / \gamma^2 ) }, $$ then their arithmetic mean $ ( X + Y ) / 2 $ again follows the Cauchy distribution of scale $\gamma$. This is a property of stable distribution and can be generalized to $n$ i.i.d. random variables. The converse is true if we are allowed to vary $n$ in the condition, as pointed out in this post, since Cauchy distribution is the unique strict stable distribution for index $ \alpha = 1 $.

I'm wondering whether the converse with only $ n = 2 $ is true. To be exact, let $\mathcal{P}$ be a probability distribution and $X$, $Y$ i.i.d. random variable following $\mathcal{P}$. We further assume the characteristic function of $\mathcal{P}$ to be symmetric, i.e. $ f (\xi) = f (-\xi) $, in order to rule out non-strict stable distributions. (This means the principle value of expectation $ \mathop{\mathbb{E}} x $ vanishses.) Given $ ( X + Y ) / 2 $ again follows $\mathcal{P}$, can we deduce that $\mathcal{P}$ is indeed the Cauchy distribution or a Dirac delta?

What I have tried so far: We have $ f (\xi) > 0 $. Take $ g (\xi) = \log f (\xi) $ and the equation is $$ g (\xi) = 2 g \left( \frac{\xi}{2} \right). $$ However, this equation is not sufficient to determine $f$ since it may not be smooth (the characteristic function of Cauchy distribution of scale $\gamma$ $$ f_{\gamma} (\xi) = \exp ( -\gamma |\xi| ) $$ is indeed not differentiable at the origin). The positive definiteness of $f$ may provide additional constraint but it is difficult to make use of.

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If you are familiar with Levy - Khinchine representation and Levy measure you can see that if $\nu$ is any symmetric Levy measure such that $\nu (2E)=2\nu (E)$ then the corresponding symmetric infinitely divisible distribution satisfies your property. So the answer is: there are lots of distributions apart from Cauchy which have this property.

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