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Some Positive Conclusions Related to the Embrechts–Goldie Conjecture

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Abstract

We give some conditions under which if an infinitely divisible distribution supported on \( [0,\infty) \) belongs to the intersection of the distribution class \( {\mathfrak{L}}(\gamma) \) for some \( \gamma\geq 0 \) and the distribution class \( {\mathfrak{OS}} \), then so does the corresponding Lévy distribution or its convolution with itself. To this end, we discuss the closure under compound convolution roots for the class and provide some types of distributions satisfying the above conditions. Therefore, this leads to some positive conclusions related to the Embrechts–Goldie conjecture in contrast to the fact that all corresponding previous results for the distribution class \( {\mathfrak{L}}(\gamma)\cap{\mathfrak{OS}} \) were negative.

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Funding

Y. Wang’s research was supported by the National Natural Science Foundation of China (no. 11071182).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Changshu Institute of Technology, Suzhou, China

    Z. Cui

  2. School of Mathematical Sciences, Soochow University, Suzhou, China

    Y. Wang & H. Xu

Authors
  1. Z. Cui
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  2. Y. Wang
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  3. H. Xu
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Corresponding author

Correspondence to Y. Wang.

Additional information

Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 1, pp. 216–231. https://doi.org/10.33048/smzh.2022.63.116

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Cui, Z., Wang, Y. & Xu, H. Some Positive Conclusions Related to the Embrechts–Goldie Conjecture. Sib Math J 63, 179–192 (2022). https://doi.org/10.1134/S0037446622010165

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  • DOI: https://doi.org/10.1134/S0037446622010165

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