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 $\bar{x}$?

Answer

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.

Attribution
Source : Link , Question Author : Molossus Spondee , Answer Author : dsaxton

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