On decomposition problem for distribution functions of class $\boldsymbol{Q}$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910764377309184 |
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| author | Khartov, A. A. |
| author_facet | Khartov, A. A. |
| contents | We consider a new class $\boldsymbol{Q}$ of distribution functions $F$ that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions $F_1$ and $F_2$ such that $F_1=F*F_2$. A distribution function of the class $\boldsymbol{Q}$ is quasi-infinitely divisible in the sense that its characteristic function admits the Lévy--Khinchine type representation with a ``signed spectral measure''. The class $\boldsymbol{Q}$, being a natural extension of the class $\boldsymbol{I}$ of infinitely divisible distribution functions, is actively studied now and it finds various applications. In 2018, Lindner, Pan and Sato formulated the open question: is it true that if $F\in\boldsymbol{Q}$ and $F=F_1*F_2$ with some distribution functions $F_1$ and $F_2$, then $F_1\in\boldsymbol{Q}$ and $F_2\in\boldsymbol{Q}$? There are some positive results under special assumptions on the type of $F$. In this paper, we answer the question in a general setting without any additional assumptions. We also consider the same question but with the stronger assumption that $F\in\boldsymbol{I}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18915 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On decomposition problem for distribution functions of class $\boldsymbol{Q}$ Khartov, A. A. Probability 60E05, 60E07, 60E10, We consider a new class $\boldsymbol{Q}$ of distribution functions $F$ that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions $F_1$ and $F_2$ such that $F_1=F*F_2$. A distribution function of the class $\boldsymbol{Q}$ is quasi-infinitely divisible in the sense that its characteristic function admits the Lévy--Khinchine type representation with a ``signed spectral measure''. The class $\boldsymbol{Q}$, being a natural extension of the class $\boldsymbol{I}$ of infinitely divisible distribution functions, is actively studied now and it finds various applications. In 2018, Lindner, Pan and Sato formulated the open question: is it true that if $F\in\boldsymbol{Q}$ and $F=F_1*F_2$ with some distribution functions $F_1$ and $F_2$, then $F_1\in\boldsymbol{Q}$ and $F_2\in\boldsymbol{Q}$? There are some positive results under special assumptions on the type of $F$. In this paper, we answer the question in a general setting without any additional assumptions. We also consider the same question but with the stronger assumption that $F\in\boldsymbol{I}$. |
| title | On decomposition problem for distribution functions of class $\boldsymbol{Q}$ |
| topic | Probability 60E05, 60E07, 60E10, |
| url | https://arxiv.org/abs/2412.18915 |