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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?

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If the mgf has a positive radius of convergence it is analytic in a vertical strip containing the imaginary axis, and the characteristic function is analytic in a horizontal strip contsining the real axis. If the characteristic function's Taylor series has a positive radius of convergence, then so does the mgf, and the above situation holds. In these cases the distribution is determined uniquely by the moments.

The problematic case is when the radii of convergence are 0.

This whole topic was hot about 100 years ago; it is now known as the Hamburger moment problem.

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  • $\begingroup$ Then, if all moments exist, there are two possible cases. i) The characteristic function is analytic at zero, and in this case actually it is analytic on the whole real line. We can conclude that the distribution is uniquely determined by the moments. ii) The characteristic function is not analytic at zero. Then we cannot deduce the determination. Am I right? Or can we say that it is not determined by the moments in this case? $\endgroup$ – Balbadak Nov 26 '18 at 15:10
  • $\begingroup$ I think you are right: in case ii, we cannot tell if the moments determine the distribution. Look at en.wikipedia.org/wiki/Carleman%27s_condition , and references. $\endgroup$ – kimchi lover Nov 26 '18 at 15:47

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