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 $\mu$ satisfies Carleman's condition, there is no other measure $\nu$ having the same moments as $\mu .$ 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:

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

Stieltjes moment problem

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

Generalized Carleman's condition

In, Nasiraee et al. showed that, despite previous assumptions, 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.

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.