# What is the distribution of sample means of a Cauchy distribution?

Typically when one takes random sample averages of a distribution (with sample size greater than 30) one obtains a normal distribution centering around the mean value. However, I heard that the Cauchy distribution has no mean value. What distribution does one obtain then when obtaining sample means of the Cauchy distribution?

Basically for a Cauchy distribution $\mu_x$ is undefined so what is $\mu_{\bar{x}}$ and what is the distribution of $\bar{x}$?

If $$X_1, \ldots, X_n$$ are i.i.d. Cauchy$$(0, 1)$$ then we can show that $$\bar{X}$$ is also Cauchy$$(0, 1)$$ using a characteristic function argument:
\begin{align} \varphi_{\bar{X}}(t) &= \text{E} \left (e^{it \bar{X}} \right ) \\ &= \text{E} \left ( \prod_{j=1}^{n} e^{it X_j / n} \right ) \\ &= \prod_{j=1}^{n} \text{E} \left ( e^{it X_j / n} \right ) \\ &= \text{E} \left (e^{it X_1 / n} \right )^n \\ &= e^{- |t|} \end{align}
which is the characteristic function of the standard Cauchy distribution. The proof for the more general Cauchy$$(\mu, \sigma)$$ case is basically identical.