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I am looking for a proof of the CLT in some simple case that does not rely on moment generating functions or characteristic functions. I think I can see how to do this in the case of Bernoulli summands, but I would be very interested to know if there is a bare hands proof out there somewhere that works for (for example) random variables taking finitely many values.

What I'm hoping for is either a reference to such a proof; or the outline of such a proof if anyone knows one. Thanks for any pointers!

anthonyquas
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    I have an intuition that there should be a fairly simple Banach-fixed-point approach to the CLT. The key idea is that the standard normal is a fixed point of the averaging process, and that averaging tends to make disparate distributions more similar. (This assumes the Banach fixed point theorem is elementary, which I think it is, from a technical point of view). – Elchanan Solomon May 20 '22 at 16:23
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    Thanks @ElchananSolomon: I believe there is a fancy method called Stein's method possibly somewhat along the lines that you suggest? – anthonyquas May 20 '22 at 16:26
  • Oh yeah, that looks like what I'm describing! I wonder if it simplifies in the case of random variables on finite sets. – Elchanan Solomon May 20 '22 at 16:32

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