Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
classical problem of moments; exponential distribution; Laplace distribution; cumulant; Marcinkiewicz theorem; analytical continuation
classical problem of moments; exponential distribution; Laplace distribution; cumulant; Marcinkiewicz theorem; analytical continuation
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 $b, (1+a-b)/2, (1-a-b)/2$.
References:
[1] Akhiezer N. I.: The classical moment problem and some related questions in analysis. Hafner Publishing Co., New York, 1965.
[2] Bieberbach L.: Analytische Forsetzung. Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), 3, Springer, Berlin, 1955 (German).
[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.
[4] Linnik J. V., Ostrovskiĭ Ĭ. V.: Decomposition of Random Variables and Vectors. Translations of Mathematical Monographs, 48, American Mathematical Society, Providence, 1977.
[5] Marcinkiewicz J.: Sur une propriété de la loi de Gauss. Math. Z. 44 (1939), no. 1, 612–618 (French). DOI 10.1007/BF01210677
[6] Shohat J. A., Tamarkin J. D.: The Problem of Moments. American Mathematical Society Mathematical Surveys, I, American Mathematical Society, New York, 1943.
[7] Szegö G.: Über Potenzreihen mit endlich vielen verschiedenen Koeffizienten. Sitzgsber. preuß. Akad. Wiss., Math.-phys. Kl. (1922), 88–91.
[8] Widder D. V.: The Laplace Transform. Princeton Mathematical Series, 6, Princeton University Press, Princeton, 1946. Zbl 0139.29504
Search
Partner of
