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.
Hamburger moment problem
For the Hamburger moment problem (the moment problem on the whole real line), the theorem states the following:
Stieltjes moment problem
For the Stieltjes moment problem, the sufficient condition for determinacy is
Generalized Carleman's condition In 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. Nasiraee et al. showed that, despite previous assumptions,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. 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
References
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.