The improvement of local limit theorem without assuming finite third moment

Jirapat Trakankitnukul; Kritsana Neammanee; Tatpon Siripraparat; Jirapat Trakankitnukul; Kritsana Neammanee; Tatpon Siripraparat

doi:10.3934/math.20261010

AIMS Mathematics

2026, Volume 11, Issue 8: 25123-25139. doi: 10.3934/math.20261010

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Research article

The improvement of local limit theorem without assuming finite third moment

1.
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
2.
Centre of Excellence in Mathematics, Ministry of Higher Education, Science, Research and Innovation, National University of Sciences, Bangkok 10400, Thailand
3.
Department of Social and Applied Science, College of Industrial Technology, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand

Received: 21 May 2026 Revised: 14 July 2026 Accepted: 27 July 2026 Published: 14 August 2026
MSC : 60F05

In this paper, we refined the local limit theorem for sums of independent integer-valued random variables under the assumption of a finite $ (2+\delta) $-moment, where $ 0 < \delta < 1 $. This class includes sums of discrete Pareto random variables, which arise naturally in various applications. Moreover, Kammoo et al. (2023) established a local limit theorem in this setting with an explicit error bound. However, the associated constants become excessively large as $ \delta \to 0 $. By combining the method of Siripraparat and Neammanee (2021) with a non-integer-order Taylor expansion, we obtained substantially sharper constants in the corresponding error bound.
Citation: Jirapat Trakankitnukul, Kritsana Neammanee, Tatpon Siripraparat. The improvement of local limit theorem without assuming finite third moment[J]. AIMS Mathematics, 2026, 11(8): 25123-25139. doi: 10.3934/math.20261010

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Abstract

In this paper, we refined the local limit theorem for sums of independent integer-valued random variables under the assumption of a finite $ (2+\delta) $-moment, where $ 0 < \delta < 1 $. This class includes sums of discrete Pareto random variables, which arise naturally in various applications. Moreover, Kammoo et al. (2023) established a local limit theorem in this setting with an explicit error bound. However, the associated constants become excessively large as $ \delta \to 0 $. By combining the method of Siripraparat and Neammanee (2021) with a non-integer-order Taylor expansion, we obtained substantially sharper constants in the corresponding error bound.

References

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