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Suppose I have a measure $\mu$ over $\mathbb R_+$ given by its moments $\mu_0,...,\mu_n$, defined as :

$$\mu_k = \int x^{k} \partial\mu(x),\; k \in 1,...,n$$ Using Faà di Bruno's formula, I can obtain the corresponding cumulants $\kappa_0,...,\kappa_n$.

Say that there exists another measure $\nu$ that happens to have the set of moments $\kappa_0,...,\kappa_n$.

Is there some work somewhere about the relationship between $\mu$ and $\nu$ ?

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    $\begingroup$ Yes, this is a very classical topic. The Nevanlinna parametrization describes all such measures (try a search for this perhaps). $\endgroup$ Aug 20, 2020 at 14:43
  • $\begingroup$ @ChristianRemling Is'nt Nevanlinna parametrization a way to solve the Hamburger moment problem ? I am not asking for a solution to my moment problem, but for a relationship between the two measures. Please tell me if i did not get what you meant. $\endgroup$
    – lrnv
    Aug 20, 2020 at 15:22

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$\newcommand\ka\kappa$$\newcommand\R{\mathbb R}$Without loss of generality, $\mu_0=1$, so that $\mu$ is a probability measure. Let $\ka$ be a measure on $\R$ with moments $\ka_0,\dots,\ka_n$ and $|\ka|:=\ka(\R)$.

Of course, only in exceptional cases (such as a case with $\mu_0=\mu_2=\mu_4=1$ and $\mu_1=0$) is a measure determined by finitely many of its moments. Therefore, usually the measure $\mu$ will not be determined by $\ka$ (nor $\ka$ will be determined by $\mu$).

However, for each measure $\ka$ with moments $\ka_0,\dots,\ka_n$ we can construct a measure $\nu$ with the same moments as $\mu$ for all moment orders from $0$ through $n$. Indeed, the accompanying (infinitely divisible) compound Poisson probability distribution $$\nu:=\nu_\ka:=e^{-|\ka|}e^{*\ka}=e^{-|\ka|}\sum_{j=0}^\infty\frac{\ka^{*j}}{j!}$$ will have the same moments as $\mu$ for all moment orders from $0$ through $n$; this is easily seen by considering the Fourier transform of $\nu_\ka$.

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  • $\begingroup$ Coming back a little late but this is still up in my mind. First thanks for the leads you gave me, very usefull stuff. However, what i wanted (without clearly knowing it at the time) was the other way around: Is there some simplification in the reconstruction of a representing measure for $\kappa$ due to the fact that $\kappa_0$,..., $\kappa_n$ are cumulants of another distribution ? $\endgroup$
    – lrnv
    Oct 5, 2020 at 13:51
  • $\begingroup$ @lrnv : I am afraid I don't understand the question in your comment. In particular, what do you mean by "a representing measure for $\kappa$"? In my answer, $\kappa$ itself is measure. $\endgroup$ Oct 5, 2020 at 18:03
  • $\begingroup$ Yes, i meant 'a representing measure for $\kappa_0,...,\kappa_n$'. My 'Big goal' is to find a measure that have moments $\kappa_0,...,\kappa_n$, and the question is: Will the fact that these moments are also cumulants of some other measure (that i completely know) help me in any way ? $\endgroup$
    – lrnv
    Oct 6, 2020 at 8:24

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