# A characterization of probability measures by f-moments

Studia Mathematica (1996)

• Volume: 118, Issue: 2, page 185-204
• ISSN: 0039-3223

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## Abstract

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Given a real-valued continuous function ƒ on the half-line [0,∞) we denote by P*(ƒ) the set of all probability measures μ on [0,∞) with finite ƒ-moments ${ʃ}_{0}^{\infty }ƒ\left(x\right){\mu }^{*n}\left(dx\right)$ (n = 1,2...). A function ƒ is said to have the identification propertyif probability measures from P*(ƒ) are uniquely determined by their ƒ-moments. A function ƒ is said to be a Bernstein function if it is infinitely differentiable on the open half-line (0,∞) and ${\left(-1\right)}^{n}{ƒ}^{\left(n+1\right)}\left(x\right)$ is completely monotone for some nonnegative integer n. The purpose of this paper is to give a necessary and sufficient condition in terms of the representing measures for Bernstein functions to have the identification property.

## How to cite

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Urbanik, K.. "A characterization of probability measures by f-moments." Studia Mathematica 118.2 (1996): 185-204. <http://eudml.org/doc/216273>.

@article{Urbanik1996,
abstract = {Given a real-valued continuous function ƒ on the half-line [0,∞) we denote by P*(ƒ) the set of all probability measures μ on [0,∞) with finite ƒ-moments $ʃ_\{0\}^\{∞\} ƒ(x)μ^\{*n\}(dx)$ (n = 1,2...). A function ƒ is said to have the identification propertyif probability measures from P*(ƒ) are uniquely determined by their ƒ-moments. A function ƒ is said to be a Bernstein function if it is infinitely differentiable on the open half-line (0,∞) and $(-1)^\{n\} ƒ^\{(n+1)\}(x)$ is completely monotone for some nonnegative integer n. The purpose of this paper is to give a necessary and sufficient condition in terms of the representing measures for Bernstein functions to have the identification property. },
author = {Urbanik, K.},
journal = {Studia Mathematica},
keywords = {Bernstein functions; Laplace transform; moments; identification properties; Laplace transform moments; identification problems},
language = {eng},
number = {2},
pages = {185-204},
title = {A characterization of probability measures by f-moments},
url = {http://eudml.org/doc/216273},
volume = {118},
year = {1996},
}

TY - JOUR
AU - Urbanik, K.
TI - A characterization of probability measures by f-moments
JO - Studia Mathematica
PY - 1996
VL - 118
IS - 2
SP - 185
EP - 204
AB - Given a real-valued continuous function ƒ on the half-line [0,∞) we denote by P*(ƒ) the set of all probability measures μ on [0,∞) with finite ƒ-moments $ʃ_{0}^{∞} ƒ(x)μ^{*n}(dx)$ (n = 1,2...). A function ƒ is said to have the identification propertyif probability measures from P*(ƒ) are uniquely determined by their ƒ-moments. A function ƒ is said to be a Bernstein function if it is infinitely differentiable on the open half-line (0,∞) and $(-1)^{n} ƒ^{(n+1)}(x)$ is completely monotone for some nonnegative integer n. The purpose of this paper is to give a necessary and sufficient condition in terms of the representing measures for Bernstein functions to have the identification property.
LA - eng
KW - Bernstein functions; Laplace transform; moments; identification properties; Laplace transform moments; identification problems
UR - http://eudml.org/doc/216273
ER -

## References

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1. [1] M. Braverman, A characterization of probability distributions by moments of sums of independent random variables, J. Theoret. Probab. 7 (1994), 187-198. Zbl0925.62056
2. [2] M. Braverman, C. L. Mallows and L. A. Shepp, A characterization of probability distributions by absolute moments of partial sums, Teor. Veroyatnost. i Primenen. 40 (1995), 270-285 (in Russian). Zbl0840.62013
3. [3] W. Feller, On Müntz' theorem and completely monotone functions, Amer. Math. Monthly 75 (1968), 342-350. Zbl0157.11502
4. [4] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, N.J., 1962. Zbl0117.34001
5. [5] S. Kaczmarz und H. Steinhaus, Theorie der Orthogonalreihen, Monograf. Mat. 6, Warszawa-Lwów, 1935. Zbl61.1119.05
6. [6] L. H. Loomis, An Introduction to Abstract Harmonic Analysis, Van Nostrand, Toronto, 1953. Zbl0052.11701
7. [7] C. Müntz, Über den Approximationssatz von Weierstrass, in: Schwarz Festschrift, Berlin, 1914, 303-312.
8. [8] M. Neupokoeva, On the reconstruction of distributions by the moments of the sums of independent random variables, in: Stability Problems for Stochastic Models, Proceedings, VNIISI, Moscow, 1989, 11-17 (in Russian).
9. [9] R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, Amer. Math. Soc., New York, 1934. Zbl0011.01601
10. [10] O. Szász, Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen, Math. Ann. 77 (1916), 482-496. Zbl46.0419.03
11. [11] K. Urbanik, Moments of sums of independent random variables, in: Stochastic Processes (Kallianpur Festschrift), Springer, New York, 1993, 321-328. Zbl0790.60016

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