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Transactions of the American Mathematical Society

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On quasi-infinitely divisible distributions


Authors: Alexander Lindner, Lei Pan and Ken-iti Sato
Journal: Trans. Amer. Math. Soc. 370 (2018), 8483-8520
MSC (2010): Primary 60E07
DOI: https://doi.org/10.1090/tran/7249
Published electronically: June 20, 2018
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Abstract: A quasi-infinitely divisible distribution on $ \mathbb{R}$ is a probability distribution whose characteristic function allows a Lévy-Khintchine type representation with a ``signed Lévy measure'', rather than a Lévy measure. Quasi-infinitely divisible distributions appear naturally in the factorization of infinitely divisible distributions. Namely, a distribution $ \mu $ is quasi-infinitely divisible if and only if there are two infinitely divisible distributions $ \mu _1$ and $ \mu _2$ such that $ \mu _1 \ast \mu = \mu _2$. The present paper studies certain properties of quasi-infinitely divisible distributions in terms of their characteristic triplet, such as properties of supports, finiteness of moments, continuity properties, and weak convergence, with various examples constructed. In particular, it is shown that the set of quasi-infinitely divisible distributions is dense in the set of all probability distributions with respect to weak convergence. Further, it is proved that a distribution concentrated on the integers is quasi-infinitely divisible if and only if its characteristic function does not have zeroes, with the use of the Wiener-Lévy theorem on absolutely convergent Fourier series. A number of fine properties of such distributions are proved based on this fact. A similar characterization is not true for nonlattice probability distributions on the line.


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Additional Information

Alexander Lindner
Affiliation: Ulm University, Institute of Mathematical Finance, Helmholtzstraße 18, 89081 Ulm, Germany
Email: alexander.lindner@uni-ulm.de

Lei Pan
Affiliation: Ulm University, Institute of Mathematical Finance, Helmholtzstraße 18, 89081 Ulm, Germany
Email: herrpan@foxmail.com

Ken-iti Sato
Affiliation: Hachiman-yama 1101-5-103, Tenpaku-ku, Nagoya, 468-0074 Japan
Email: ken-iti.sato@nifty.ne.jp

DOI: https://doi.org/10.1090/tran/7249
Received by editor(s): January 8, 2017
Received by editor(s) in revised form: March 29, 2017
Published electronically: June 20, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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