Question about Humburger moment problem and Characteristic function.

Question

Let $\phi(t)$ be some a contionous infinitly differentialbe function such that $\phi(0)=1$ and $\phi(t)$ is symmmetric.
Let \begin{align} m_{2n} =i^{-2n} \phi^{n}(0) \end{align}

Suppose, that \begin{align} \sum_{n=0}^\infty m_{2n}^{-\frac{1}{2n}}=\infty \end{align}

That is $m_n$'s satisfy the Carleman's condition.

Does this imply that $\phi(t)$ is a characteristic function of some distribution?

Thanks.

Answer 1

I don't know neither Humburger, nor Humberger problem. If you're speaking about Hamburger problem, then no, this is not assumed. Carleman's condition supplies that there is at most one solution to the moment problem. The existence of solution is provided by another condition (in fact, a criterion) that the matrix $(m_{i+j})$ is positive definite. Returning to your particular question, even if $(m_n)$ is a valid sequence of moments, $\phi$ does not need to be a characteristic function: you impose conditions in the point $0$ only, but elsewhere $\phi$ can misbehave arbitrarily. There can be some positive answers. Say, if $\phi$ is analytic, I think it can be verified (and I welcome you to) from the criterion I cite that $\phi$ is a characteristic function.