Open Access
2023 Subexponentialiy of densities of infinitely divisible distributions
Muneya Matsui
Author Affiliations +
Electron. J. Probab. 28: 1-29 (2023). DOI: 10.1214/23-EJP928


We show the equivalence of three properties for an infinitely divisible distribution: the subexponentiality of the density, the subexponentiality of the density of its Lévy measure and the tail equivalence between the density and its Lévy measure density, under monotonic-type assumptions on the Lévy measure density. The key assumption is that tail of the Lévy measure density is asymptotic to a non-increasing function or is almost decreasing. Our conditions are natural and cover a rather wide class of infinitely divisible distributions. Several significant properties for analyzing the subexponentiality of densities have been derived such as closure properties of [ convolution, convolution roots and asymptotic equivalence ] and the factorization property. Moreover, we illustrate that the results are applicable for developing the statistical inference of subexponential infinitely divisible distributions which are absolutely continuous.

Funding Statement

The author’s research is partly supported by the JSPS Grant-in-Aid for Scientific Research C (19K11868).


The earlier version of results in this paper has been presented at the annual workshop “Infinitely divisible processes and related topics” held in Nov. 2021. The results have been significantly improved after the workshop and the author acknowledges the comments and the hosts in the workshop. The author is grateful to Toshiro Watanabe for careful reading and all comments and discussions about subexponentiality. Particularly, his suggestion of the relation between the local subexponentiality and the topic yielded substantial improvement of the main theorem.


Download Citation

Muneya Matsui. "Subexponentialiy of densities of infinitely divisible distributions." Electron. J. Probab. 28 1 - 29, 2023.


Received: 2 October 2022; Accepted: 2 March 2023; Published: 2023
First available in Project Euclid: 13 March 2023

MathSciNet: MR4559328
zbMATH: 1517.60025
MathSciNet: MR4529085
Digital Object Identifier: 10.1214/23-EJP928

Primary: 60E07 , 60G70 , 62F12

Keywords: almost decreasing , asymptotic to a non-increasing function , Infinite divisibility , Lévy measure , long-tailedness , subexponential density , tail equivalence

Vol.28 • 2023
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