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Variations of central limit theorems and Stirling numbers of the first kind
Author(s):
1. Bernhard Heim: University of Cologne, Weyertal 86–90, 50931, Germany ;Lehrstuhl A fur Mathematik, RWTH Aachen University, 52056 Aachen, Germany
2. Markus Neuhauser: Kutaisi International University, 5/7, Youth Avenue, 4600 Kutaisi, Georgia;Lehrstuhl A f ¨ ur Mathematik, RWTH Aachen University, 52056 Aachen, Germany
Abstract:
We construct a new parametrization of double sequences fAn;k(s)gn;k between An;k(0) = nk 11 and An;k(1) = n1! nk , where nk are the unsigned Stirling numbers of the first kind. For each s, we prove a central limit theorem and a local limit theorem. This extends the de Moivre-Laplace central limit theorem and Goncharov's result that unsigned Stirling numbers of the first kind are asymptotically normal. We also provide several applications.
Page(s): 54-61
Published: Journal: Discrete Mathematics Letters, Volume: 12, Issue: 0, Year: 2023
Keywords:
local limit theorem , central limit theorem , singularity analysis , probabilistic number theory
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