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Closure under infinitely divisible distribution roots and the Embrechts–Goldie conjecture

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Abstract

We show that the distribution class ℒ(γ) \ 𝒪𝒮 is not closed under infinitely divisible distribution roots for γ > 0, that is, we provide some infinitely divisible distributions belonging to the class, whereas the corresponding Lévy distributions do not. In fact, one part of these Lévy distributions belonging to the class 𝒪ℒ\ℒ(γ) have different properties, and the other parts even do not belong to the class 𝒪ℒ. Therefore, combining with the existing related results, we give a completely negative conclusion for the subject and Embrechts–Goldie conjecture. Then we discuss some interesting issues related to the results of this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Soochow University, Suzhou, 215006, China

    Hui Xu, Yuebao Wang & Dongya Cheng

  2. School of Sciences, Nantong University, Nantong, 226019, China

    Changjun Yu

Authors
  1. Hui Xu
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  2. Changjun Yu
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  3. Yuebao Wang
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  4. Dongya Cheng
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Correspondence to Yuebao Wang.

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Yuebao Wang was supported by National Natural Science Foundation of China (No. 11071182).

Dongya Cheng was supported by National Natural Science Foundation of China (No. 11401415), Tian Yuan foundation (No. 11426139), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 13KJB110025), and Postdoctoral Research Program of Jiangsu Province of China (No. 1402111C).

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Xu, H., Yu, C., Wang, Y. et al. Closure under infinitely divisible distribution roots and the Embrechts–Goldie conjecture. Lith Math J 64, 101–114 (2024). https://doi.org/10.1007/s10986-024-09620-8

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  • DOI: https://doi.org/10.1007/s10986-024-09620-8

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