The cat in the lab with the bow tie is my spirit animal.
For the alpha nerds:
My favorite free statistics notes (from a class I was once a TA for):
http://www.math.caltech.edu/~2016-17/2term/ma003/
You need to make some small assumptions on the mean and variance of your random variable for the central limit theorem to work.
The central limit theorem, as I stated it, deals with the probability density/mass functions. There is a version of the theorem like this under some small assumptions of the moments of the distribution. However, the actual central limit theorem deals with the convergence w.r.t the cumulative distribution functions.
If you are interested in the rate of convergence in the central limit theorem, look up the Berry Essen theorem.
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