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Carleman's condition

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In mathematics, particularly, in analysis, Carleman's condition gives a sufficient condition for the determinacy of the moment problem. That is, if a measure satisfies Carleman's condition, there is no other measure having the same moments as The condition was discovered by Torsten Carleman in 1922.[1]

Hamburger moment problem

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For the Hamburger moment problem (the moment problem on the whole real line), the theorem states the following:

Let be a measure on such that all the moments are finite. If then the moment problem for is determinate; that is, is the only measure on with as its sequence of moments.

Stieltjes moment problem

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For the Stieltjes moment problem, the sufficient condition for determinacy is


Generalized Carleman's condition

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In,[2] Nasiraee et al. showed that, despite previous assumptions,[3] when the integrand is an arbitrary function, Carleman's condition is not sufficient, as demonstrated by a counter-example. In fact, the example violates the bijection, i.e. determinacy, property in the probability sum theorem. When the integrand is an arbitrary function, they further establish a sufficient condition for the determinacy of the moment problem, referred to as the generalized Carleman's condition.

Notes

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  1. ^ Akhiezer (1965)
  2. ^ M. Nasiraee, Jav. Kazemitabar and Jal. Kazemitabar, "The Bijection Property in the Law of Total Probability and Its Application in Communication Theory," in IEEE Communications Letters, doi: 10.1109/LCOMM.2024.3447352.
  3. ^ S. S. Shamai, “Capacity of a pulse amplitude modulated direct detection photon channel,” IEE Proceedings I (Communications, Speech and Vision), vol. 137, no. 6, pp. 424–430, Dec. 1990.

References

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  • Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd.
  • Chapter 3.3, Durrett, Richard. Probability: Theory and Examples. 5th ed. Cambridge Series in Statistical and Probabilistic Mathematics 49. Cambridge ; New York, NY: Cambridge University Press, 2019.