# Mean of two i.i.d. random variables follows the same distribution, Cauchy distribution?

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.

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.