Carleman's condition explained

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

Stieltjes moment problem

For the Stieltjes moment problem, the sufficient condition for determinacy is$$\backslash sum\_^\backslash infty\; m\_^\; =\; +\; \backslash infty.$$

References

• Book: Akhiezer, N. I. . The Classical Moment Problem and Some Related Questions in Analysis . Oliver & Boyd . 1965.
• 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.