Variations of central limit theorems and Stirling numbers of the first kind | [Discrete Mathematics Letters • 2023]

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

DOI: 10.47443/dml.2022.183

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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