Carleman's condition

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

Hamburger moment problem

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

Let ${\displaystyle \mu }$ be a measure on ${\displaystyle \mathbb {R} }$ such that all the moments $${\displaystyle m_{n}=\int _{-\infty }^{+\infty }x^{n}\,d\mu (x)~,\quad n=0,1,2,\cdots }$$ are finite. If $${\displaystyle \sum _{n=1}^{\infty }m_{2n}^{-{\frac {1}{2n}}}=+\infty ,}$$ then the moment problem for ${\displaystyle (m_{n})}$ is determinate; that is, ${\displaystyle \mu }$ is the only measure on ${\displaystyle \mathbb {R} }$ with ${\displaystyle (m_{n})}$ as its sequence of moments.

Stieltjes moment problem

For the Stieltjes moment problem, the sufficient condition for determinacy is $${\displaystyle \sum _{n=1}^{\infty }m_{n}^{-{\frac {1}{2n}}}=+\infty .}$$

Generalized Carleman's condition

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

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

• 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.