Posted: Oct. 25, 2025
Hello Doron,
I am Jordan Stoyanov, retired probabilist, still enjoying Math, reading interesting papers.
Some days ago I saw your arXiv "evil numbers' paper. I saved it, as a valuable addition to my collection of math papers with short and original title.
Today I spent time, and it seems I understood your idea and conclusion.
My only comment is about concluding the asymptotic normality of that location random variable. Moment determinacy of probability distributions in terms of their moments is one of my favorite topics over the last decades. I have a series of papers, by myself and joint with friends, also there is a chapter in my book "Counterexamples in Probability. 3rd ed.", Dover 2013.
If for you available are only the kurtosis and the skewness, even say, very close to 0 and 3, you CANNOT conclude that the distribution is close to the Normal, N(0, 1). In fact, there are infinitely many distributions, absolutely continuous and discrete, with the same kurtosis and skewness as N (0, 1).
Now, if you have convergence of ALL moments, to the moments of N(0,1), you can claim that the limit distribution is N(0,1). The uniqueness of N in terms of the moments is essential. This goes back to Chebyshev and Markov.
Even if you check that the moments of order UP TO n (for you n = 16) are close to those of N, it is NOT ENOUGH to conclude the asymptotic normality, even if n is large.
Maybe in your case there is a way to show theoretically convergence of all moments to those of N.
There is another idea, to use the cumulant function, the log of the m.g.f. It is known that for N(0,1), the cumulants are k_1 = 0, k_2 = 1, k_j = 0 for all j = 3, 4,.... (\infty). Thus, if you manage to work with the cumulants of the location r.v., and show convergence to 0, 1, 0, 0, ... (zeros), you are done.
Anyway, your work is interesting, and I hope you will have what more to say to the math audience.
Best regards, Jordan Stoyanov (Email: stoyanovj@gmail.com )