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

proposition:

If is a sequence of independent Cauchy distributed random variables,

then has a Cauchy distribution.

Lemma: If is Cauchy distributed, then

Proof:

We want to compute the characteristic function of and compare it to the characteristic function of a Cauchy distributed random variable…

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