# Cauchy Distribution

In my previous post, I gave a demonstration of the Central Limit Theorem
for any data having finite mean and finite variance. And as the Cauchy
distribution will show, this last requirement can’t be relaxed.

Definition: a random variable $X$ is Cauchy distributed if the density of $X$ is given by $F(x) = \frac{1}{\pi (1+x^2)}$

It’s trivial to show that the expectation and variance for a Cauchy distributed
random variable are both infinite so we may now proceed to make the following
proposition:

If $\{X_{n}\}$  is a sequence of independent Cauchy distributed random variables,
then $Y_{n} = \frac{1}{n}\sum_{i=1}^{n} X_{i}$ has a Cauchy distribution.

Lemma: If $X$ is Cauchy distributed, then $\varphi_{X}(t) = e^{-|t|}$

Proof:
We want to compute the characteristic function of $Y_{n}$ and compare it to the characteristic function of a Cauchy distributed random variable…

$\varphi_{Y_{n}}(t) = \prod_{n=1}^n\varphi_{\frac{X_{i}}{n}}(t) = \prod_{n=1}^n\varphi_{X_{i}}(t/n)=(\varphi_{X_{1}}(t/n))^n=(e^{-\frac{|t|}{n}})^n=e^{-|t|}$