## Limit theorems for the least common multiple of a random set of integers

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- by Gerold Alsmeyer, Zakhar Kabluchko and Alexander Marynych PDF
- Trans. Amer. Math. Soc.
**372**(2019), 4585-4603 Request permission

## Abstract:

Let $L_{n}$ be the least common multiple of a random set of integers obtained from $\{1,\ldots ,n\}$ by retaining each element with probability $\theta \in (0,1)$ independently of the others. We prove that the process $(\log L_{\lfloor nt\rfloor })_{t\in [0,1]}$, after centering and normalization, converges weakly to a certain Gaussian process that is not Brownian motion. Further results include a strong law of large numbers for $\log L_{n}$ as well as Poisson limit theorems in regimes when $\theta$ depends on $n$ in an appropriate way.## References

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## Additional Information

**Gerold Alsmeyer**- Affiliation: Institute of Mathematical Stochastics, Department of Mathematics and Computer Science, University of Münster, Orléans-Ring 10, D-48149, Münster, Germany
- MR Author ID: 25115
- Email: gerolda@uni-muenster.de
**Zakhar Kabluchko**- Affiliation: Institute of Mathematical Stochastics, Department of Mathematics and Computer Science, University of Münster, Orléans-Ring 10, D-48149, Münster, Germany
- MR Author ID: 696619
- ORCID: 0000-0001-8483-3373
- Email: zakhar.kabluchko@uni-muenster.de
**Alexander Marynych**- Affiliation: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine
- MR Author ID: 848771
- Email: marynych@unicyb.kiev.ua
- Received by editor(s): January 26, 2018
- Published electronically: July 2, 2019
- Additional Notes: The first and second authors were partially supported by the Deutsche Forschungsgemeinschaft (SFB 878)

The third author was partially supported by the Alexander von Humboldt Foundation - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 4585-4603 - MSC (2010): Primary 60F05; Secondary 11N37, 60F15
- DOI: https://doi.org/10.1090/tran/7871
- MathSciNet review: 4009436