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|}

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