Carleman's condition

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.

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

$${\displaystyle \sum _{n=1}^{\infty }m_{n}^{-{\frac {1}{2n}}}=+\infty .}$$

• 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.