Question on Cauchy-distribution. Question: Let $X, Y$ be independent and Cauchy-distributed and define $Z:=$ $X+Y$. Show that $Z / 2=\frac{X+Y}{2}$ is Cauchy-distributed.

My attempt: I think that we can find the density of $Z$ first. For any fixed $a \in \mathbb{R}$, the derivative of the function
$$
h(y):= \begin{cases}\frac{1}{4 a+a^3}\left(\log \left(\left(1+y^2\right) /\left(1+(a-y)^2\right)\right)+\operatorname{aatan}(y)-a \operatorname{atan}(a-y)\right) & a \neq 0, \\ \frac{1}{4}\left(\frac{2 y}{1+y^2}+2 \operatorname{atan}(y)\right) & a=0,\end{cases}
$$
is given by
$$
h^{\prime}(y)=\frac{1}{1+(a-y)^2} \frac{1}{1+y^2} .
$$
where atan $: \mathbb{R} \rightarrow(-\pi / 2, \pi / 2)$ is the inverse of the tangent function which satisfies $h(\infty)=\frac{\pi}{4+a^2}$ and $h(-\infty)=\frac{-\pi}{4+a^2}$ where these expression are meant as the respective limits.
And if $X_1, \ldots, X_n$ are independent Cauchy-distributed random variables, then the average $\frac{1}{n} \sum_{k=1}^n X_k$ is Cauchy-distributed.
But I'm not sure how to solve this question. Help would be appreciated. Thank you.
 A: As you only consider linear combinations of Cauchy distributions, the easiest way to demonstrate that it is also Cauchy uses their characteristic functions.
The characteristic function of a Cauchy law $X$ centered at $a$ with scale parameter $\gamma$ is given by $\phi_X(t) = \mathbb{E}[e^{itX}] = e^{ita-\gamma|t|}$. Given properties of the characteristic function, one finds easily the characteristic function of the linear combination $Z = \sum_k \lambda_kX_k$, where $X_k\sim\mathrm{Cauchy}(a_k,\gamma_k)$, as follows :
$$
\phi_Z(t) = \prod_k\phi_{X_k}(\lambda_kt) = \prod_k e^{it\lambda_ka_k-\gamma_k|\lambda_kt|} = \exp\left(\sum_k it\lambda_ka_k-\gamma_k|\lambda_kt|\right) = e^{ita-\gamma|t|}
$$
with
$$
a = \sum_k\lambda_ka_k \verb+  +\mathrm{and}\verb+  + \gamma = \sum_k|\lambda_k|\gamma_k
$$
whence one concludes that $Z\sim\mathrm{Cauchy}(a,\gamma)$ is also Cauchy.
Note that distributions whose nature stays the same through linear combinations are called "stable" and play an important role in generalized versions of the central limit theorem.
