Article
| Title: | On an arithmetical property of moments and cumulants (English) |
| Author: | Kakosyan, Ashot V. |
| Author: | Klebanov, Lev B. |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 66 |
| Issue: | 1 |
| Year: | 2025 |
| Pages: | 71-79 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | This paper examines the moments of probability distributions, presenting new theorems and their implications. The main result of the paper is the following. Let a nondegenerate distribution have finite moments $\mu_k$ of all orders $k=0,1,2,\ldots$ Then the sequence $\{\mu_k/k!\colon k=0,1,2,\ldots\}$ 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+ {\rm i}at+bt^2}{1+t^2}, \qquad \text{where } b\geq 0, 1-a-b \geq 0, 1+a-b \geq 0. \end{equation*} Corresponding distribution is mixture of an atom at zero, exponential distribution on positive semiaxis and exponential distribution on negative semiaxis with weights $b, (1+a-b)/2, (1-a-b)/2$. (English) |
| Keyword: | classical problem of moments |
| Keyword: | exponential distribution |
| Keyword: | Laplace distribution |
| Keyword: | cumulant |
| Keyword: | Marcinkiewicz theorem |
| Keyword: | analytical continuation |
| MSC: | 30B40 |
| MSC: | 30E05 |
| MSC: | 44A60 |
| MSC: | 60E05 |
| MSC: | 60E10 |
| DOI: | 10.14712/1213-7243.2026.002 |
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| Date available: | 2026-05-15T05:56:20Z |
| Last updated: | 2026-05-18 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153614 |
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| Reference: | [1] Akhiezer N. I.: The classical moment problem and some related questions in analysis.Hafner Publishing Co., New York, 1965. |
| Reference: | [2] Bieberbach L.: Analytische Forsetzung.Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), 3, Springer, Berlin, 1955 (German). |
| Reference: | [3] Klebanov L. B., Melamed J. A.: Generalized Marcinkiewicz's theorem and asymptotic expansions in the central limit theorem.New Trends in Probability and Statistics, 1, Bakuriani, 1990, VSP, Utrecht, 1991, 30–37. |
| Reference: | [4] Linnik J. V., Ostrovskiĭ Ĭ. V.: Decomposition of Random Variables and Vectors.Translations of Mathematical Monographs, 48, American Mathematical Society, Providence, 1977. |
| Reference: | [5] Marcinkiewicz J.: Sur une propriété de la loi de Gauss.Math. Z. 44 (1939), no. 1, 612–618 (French). 10.1007/BF01210677 |
| Reference: | [6] Shohat J. A., Tamarkin J. D.: The Problem of Moments.American Mathematical Society Mathematical Surveys, I, American Mathematical Society, New York, 1943. |
| Reference: | [7] Szegö G.: Über Potenzreihen mit endlich vielen verschiedenen Koeffizienten.Sitzgsber. preuß. Akad. Wiss., Math.-phys. Kl. (1922), 88–91. |
| Reference: | [8] Widder D. V.: The Laplace Transform.Princeton Mathematical Series, 6, Princeton University Press, Princeton, 1946. Zbl 0139.29504 |
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