On an arithmetical property of moments and cumulants
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909139858358272 |
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| author | Kakosyan, Ashot V. Klebanov, Lev B. |
| author_facet | Kakosyan, Ashot V. Klebanov, Lev B. |
| contents | The main result of the paper is the following.
Let a non-degenerate distribution have finite moments $μ_k$ of all orders $k=0,1,2,\ldots$. Then the sequence $\{μ_k/k!, \; k=0,1,2,\ldots\}$ either contains infinitely many different terms or at most three. In the latter case, this sequence has the form $\{1,a,1-b,a,1-b,a,1-b, \ldots\}$ and corresponds to a distribution with the characteristic function \begin{equation*}\label{ eq0}
f(t)=\frac{1+iat+bt^2}{1+t^2}, \quad \text{where} \;\; b\geq 0,\; \frac{1-a-b}{2}\geq 0,\; \frac{1+a-b}{2}\geq 0.
\end{equation*}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_11906 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On an arithmetical property of moments and cumulants Kakosyan, Ashot V. Klebanov, Lev B. Probability ACM-class: 60E05, 60E10, 44A60, 30E05, 30B40 The main result of the paper is the following. Let a non-degenerate distribution have finite moments $μ_k$ of all orders $k=0,1,2,\ldots$. Then the sequence $\{μ_k/k!, \; k=0,1,2,\ldots\}$ either contains infinitely many different terms or at most three. In the latter case, this sequence has the form $\{1,a,1-b,a,1-b,a,1-b, \ldots\}$ and corresponds to a distribution with the characteristic function \begin{equation*}\label{ eq0} f(t)=\frac{1+iat+bt^2}{1+t^2}, \quad \text{where} \;\; b\geq 0,\; \frac{1-a-b}{2}\geq 0,\; \frac{1+a-b}{2}\geq 0. \end{equation*}. |
| title | On an arithmetical property of moments and cumulants |
| topic | Probability ACM-class: 60E05, 60E10, 44A60, 30E05, 30B40 |
| url | https://arxiv.org/abs/2403.11906 |