# characteristic function and distribution completely determined by moments

Let $$X$$ be a real valued random variable. My textbook states that if the moment generating function $$E[e^{sX}]$$ is finite in a neighborhood of zero, the distribution of $$X$$ is determined completely by the moments. However, I cannot find a similar statement with the characteristic function $$E[e^{i t X}]$$. So I tried to deduce one.

Let $$X_1$$ and $$X_2$$ have the same moments of all orders.

1. Then their characteristic functions $$\phi_{X_1}$$ and $$\phi_{X_2}$$ are infinitely differentiable at $$0$$ and the derivatives have the same value at $$0$$. $$\phi_{X_1}$$ and $$\phi_{X_2}$$ have the same Taylor series expansion at $$0$$.
2. If the radius of convergence of the series is infinite, $$\phi_{X_1}=\phi_{X_2}$$. A characteristic function uniquely determines the distribution. So we conclude that $$X_1$$ and $$X_2$$ have the same distribution.
3. If the radius of convergence is zero, we cannot say that $$\phi_{X_1}=\phi_{X_2}$$ nor the same distribution.
4. If the radius of convergence is positive but finite, we cannot say that $$\phi_{X_1}(s)=\phi_{X_2}(s)$$ for $$s$$ beyond the radius of convergence. If we consider analytic extensions of $$\phi_{X_1}$$ and $$\phi_{X_2}$$ on the complex domain, they have singularities somewhere but it may be possible that $$\phi_{X_1} = \phi_{X_2}$$ for whole real line and the singularities could give helpful information on this.

Q1: From the above argument (2), if a probability distribution has all moments and its characteristic function is analytic in the whole real line, the distribution is completely determined by the moments. Is this correct?

Q2: Are there some useful theorems considering the analytic extension of a characteristic function on complex plane, related to the determination by moments?