# 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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topUrbanik, 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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- [7] C. Müntz, Über den Approximationssatz von Weierstrass, in: Schwarz Festschrift, Berlin, 1914, 303-312.
- [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] R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, Amer. Math. Soc., New York, 1934. Zbl0011.01601
- [10] O. Szász, Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen, Math. Ann. 77 (1916), 482-496. Zbl46.0419.03
- [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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