## Quasi-infinite divisibility of a class of distributions with discrete part

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- by David Berger and Merve Kutlu
- Proc. Amer. Math. Soc.
**151**(2023), 2211-2224 - DOI: https://doi.org/10.1090/proc/16312
- Published electronically: February 10, 2023
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## Abstract:

We consider distributions on $\mathbb {R}$ that can be written as the sum of a non-zero discrete distribution and an absolutely continuous distribution. We show that such a distribution is quasi-infinitely divisible if and only if its characteristic function is bounded away from zero, thus giving a new class of quasi-infinitely divisible distributions. Moreover, for this class of distributions we characterize the existence of the $g$-moment for certain functions $g$.## References

- I. A. Alexeev and A. A. Khartov,
*Spectral representations of characteristic functions of discrete probability laws*, Preprint, arXiv:2101.06038, 2021. - Takahiro Aoyama and Takashi Nakamura,
*Behaviors of multivariable finite Euler products in probabilistic view*, Math. Nachr.**286**(2013), no. 17-18, 1691–1700. MR**3145164**, DOI 10.1002/mana.201200151 - Radu Balan and Ilya Krishtal,
*An almost periodic noncommutative Wiener’s lemma*, J. Math. Anal. Appl.**370**(2010), no. 2, 339–349. MR**2651657**, DOI 10.1016/j.jmaa.2010.04.053 - David Berger,
*On quasi-infinitely divisible distributions with a point mass*, Math. Nachr.**292**(2019), no. 8, 1674–1684. MR**3994295**, DOI 10.1002/mana.201800073 - David Berger, Franziska Kühn, and René L. Schilling,
*Lévy processes, generalized moments and uniform integrability*, Probab. Math. Statist.**42**(2022), no. 1, 109–131. MR**4490672** - Loïc Chaumont and Andreas E. Kyprianou,
*A lifetime of excursions through random walks and Lévy processes*, A lifetime of excursions through random walks and Lévy processes—a volume in honour of Ron Doney’s 80th birthday, Progr. Probab., vol. 78, Birkhäuser/Springer, Cham, [2021] ©2021, pp. 1–11. MR**4425782**, DOI 10.1007/978-3-030-83309-1_{1} - David Berger and Alexander Lindner,
*A Cramér-Wold device for infinite divisibility of $\Bbb Z^d$-valued distributions*, Bernoulli**28**(2022), no. 2, 1276–1283. MR**4388938**, DOI 10.3150/21-bej1386 - Hassan Chhaiba, Nizar Demni, and Zouhair Mouayn,
*Analysis of generalized negative binomial distributions attached to hyperbolic Landau levels*, J. Math. Phys.**57**(2016), no. 7, 072103, 14. MR**3522606**, DOI 10.1063/1.4958724 - Constantin Corduneanu,
*Almost periodic oscillations and waves*, Springer, New York, 2009. MR**2460203**, DOI 10.1007/978-0-387-09819-7 - R. Cuppens,
*Decomposition of multivariate probabilities*, Academic Press, New York, 1975. - Nizar Demni and Zouhair Mouayn,
*Analysis of generalized Poisson distributions associated with higher Landau levels*, Infin. Dimens. Anal. Quantum Probab. Relat. Top.**18**(2015), no. 4, 1550028, 13. MR**3447229**, DOI 10.1142/S0219025715500289 - Lawrence C. Evans,
*Partial differential equations*, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR**2597943**, DOI 10.1090/gsm/019 - I. Gelfand, D. Raikov, and G. Shilov,
*Commutative normed rings*, Chelsea Publishing Co., New York, 1964. Translated from the Russian, with a supplementary chapter. MR**205105** - K. Gröchenig,
*Wiener’s lemma: theme and variations. An introduction to spectral invariance and its applications*, Four Short Courses on Harmonic Analysis, Birkhäuser, Boston, 2010, pp. 175–234. - A. A. Khartov,
*A criterion of quasi-infinite divisibility for discrete laws*, Statist. Probab. Lett.**185**(2022), Paper No. 109436, 4. MR**4393935**, DOI 10.1016/j.spl.2022.109436 - M. G. Krein,
*Integral equations on a half-line with kernel depending upon the difference of the arguments*, Trans. Amer. Math. Soc.**22**(1962), 163–288. - B. M. Levitan and V. V. Zhikov,
*Almost periodic functions and differential equations*, Cambridge University Press, Cambridge-New York, 1982. Translated from the Russian by L. W. Longdon. MR**690064** - A. Lindner, L. Pan, and K. Sato,
*On quasi-infinitely divisible distributions*, Trans. Amer. Math. Soc.**370**(2018), 8483–8520. - Takashi Nakamura,
*A complete Riemann zeta distribution and the Riemann hypothesis*, Bernoulli**21**(2015), no. 1, 604–617. MR**3322332**, DOI 10.3150/13-BEJ581 - Ken-iti Sato,
*Lévy processes and infinitely divisible distributions*, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 2013. Translated from the 1990 Japanese original; Revised edition of the 1999 English translation. MR**3185174** - Huiming Zhang, Yunxiao Liu, and Bo Li,
*Notes on discrete compound Poisson model with applications to risk theory*, Insurance Math. Econom.**59**(2014), 325–336. MR**3283233**, DOI 10.1016/j.insmatheco.2014.09.012

## Bibliographic Information

**David Berger**- Affiliation: Technische Universität Dresden, Institut für Mathematische Stochastik, 01062 Dresden, Germany
- MR Author ID: 1398889
- Email: david.berger2@tu-dresden.de
**Merve Kutlu**- Affiliation: Universität Ulm, Institut für Finanzmathematik, 89081 Ulm, Germany
- MR Author ID: 1441249
- Email: merve.kutlu@live.de
- Received by editor(s): April 21, 2022
- Received by editor(s) in revised form: September 13, 2022
- Published electronically: February 10, 2023
- Additional Notes: This work was financially supported through the DFG-NCN Beethoven Classic 3 project SCHI419/11-1 & NCN 2018/31/G/ST1/02252.
- Communicated by: Amarjit Budhiraja
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**151**(2023), 2211-2224 - MSC (2020): Primary 60E07, 60E10; Secondary 60E05
- DOI: https://doi.org/10.1090/proc/16312
- MathSciNet review: 4556212