By the central limit theorem (CLT), I mean the Lindeberg-Lévy CLT that says if $X\_1,X\_2,\ldots$ are i.i.d. random variables with $\mathbf{E}[X\_1] = 0$ and $\mathbf{E}[X\_1^2] = 1$, then
$$ \frac{X\_1+\cdots+X\_n}{\sqrt{n}}$$
converges in distribution to the standard normal random variable.


The de Moivre-Laplace theorem, which is historically the first instance of the CLT, is essentially the special case of CLT where $X\_1$ has only two possible values. In other words, it states that in certain cases, binomial distributions can be approximated by normal distributions. The de Moivre-Laplace theorem can be proved by direct computation, where one uses Stirling's formula.


I found a (probably) new proof of the CLT that derives it from de Moivre-Laplace theorem. The proof is fairly elementary and avoids using the characteristic function. I wrote a 5-page note on it and submitted to the American Mathematical Monthly. However, they rejected it, saying that "the proof in this article is new to the Board, and "clever". BUT the argument gives neither conceptual insight into why the result is true, nor potential for generalization (the remarkable feature of the CLT is that in fact it is true in much greater generality). It's "just a proof" and we do not think our Monthly readers will find it interesting."


I mostly agree to what they said, but still think this proof is worth putting out somewhere. Could anyone suggest journals where it would be appropriate to submit this kind of notes? I should say that this note is not elementary to the degree that undergraduate students would easily read it.



Where should I submit a derivation of the CLT from the de Moivre-Laplace theorem?
I would be personally interested to see your proof. You should compare it to the Lindeberg proof that also does not use characteristic functions but replaces the variables one by one by Gaussians. You can find an exposition of this proof in the book [1] or in [Chin - A Short and Elementary Proof of the Central Limit Theorem by Individual Swapping](https://arxiv.org/abs/2106.00871).


Regarding your query: [Expositiones Mathematicae](https://www.journals.elsevier.com/expositiones-mathematicae) might be a relevant journal. See also the long list and discussion in [Which journals publish expository work?](https://mathoverflow.net/questions/15366/which-journals-publish-expository-work?rq=1)


[1] Breiman, Leo. "[Probability](https://doi.org/10.1137/1.9781611971286), Classics in Applied Mathematics, vol. 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992." Corrected reprint of the 1968 original. MR1 163370.


