The Automatic Central Limit Theorems Generator (and Much More!)

By Doron Zeilberger

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[Appeared in: "Advances in Combinatorial Mathematics: Proceedings of the Waterloo Workshop in Computer Algebra 2008 in honor of Georgy P. Egorychev", chapter 8, pp. 165-174, (I.Kotsireas, E.Zima, eds. Springer Verlag, 2009.)]
Written: Dec. 4, 2008.

Dedicated to Georgy Petrovich EGORYCHEV, on his 70th Birthday.

Back in the late-seventies and early eighties, Georgy Egorychev built a very important bridge between the discrete and the continuous, by reducing many combinatorial sums and multisums to contour integrals and multi-integrals. This is yet another bridge between the Discrete and the Continuous, this time in Probability Theory.
Added Dec. 20, 2010: In May 2009 I gave a short talk about this work, at the CANADAM 2009 conference, that has just been uploaded to YouTube. Here is it is:
part1   part2   part3   .

Maple Packages

Important: This article is accompanied by Maple packages

Sample Input and Output for CLT

To get a refined version of the Central Limit Theorem, the input gives the output.

Sample Input and Output for AsymptoticMoments

  • To get the aympototics for the moments of the Binomial Distribution, the mahonian statistics (inversions on permutations) and the q-Catalan distribution, the input gives the output.

  • To get the asymptotics of the even moments (the odd moments are zero) for rolling a fair k-faced die for k between 2 and 6, the input gives the output.

  • To get the asymptotics for the even moments (the odd moments are zero) for plane partitions bounded in an n by n by k box for k between 1 and 4, the input gives the output.
Added March 27, 2009: Read Herb Wilf's insightful comment
Added June 8, 2009: Georgy Petrovich EGORYCHEV invited me to visit Krasnoyarsk, during May 10-16, 2009, and I had a great time, thanks to the great hospitality of Georgy, his wife Zina, and his amazing, computer-whiz, student, Maxim DAVLETSHIN. Maxim took lots of pictures Thanks Maxim!
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