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 is Cauchy distributed if the density of is given by
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
If is a sequence of independent Cauchy distributed random variables,
then has a Cauchy distribution.
Lemma: If is Cauchy distributed, then
We want to compute the characteristic function of and compare it to the characteristic function of a Cauchy distributed random variable…