For a random variable $X$, the cumulant generating function $CGF_X$ is defined as $CGF_X(t)=\log Ee^{tX}$, and the nth cumulant $k_n(X)$ is defined as the coefficient of $t^n/n!$ in the corresponding power series. The cumulant $k_n$ has the following properties:

  1. $k_n(X+Y)=k_n(X)+k_n(Y)$ if X and Y are independent (additivity)
  2. $k_n(cX)=c^nk_n(X)$ for any scalar $c$ (homogeneity)
  3. $k_n(X)=p_n(EX, EX^2,\dots, EX^n)$ where $p_n$ is a universal polynomial (i.e. does not depend on X)

Now suppose I have some other function $k'$ that satisfies properties 1-3. Is k' necessarily a scalar multiple of $k_n$?

Motivation: The higher order cumulants can be somewhat mysterious-having a characterization like the above would make them seem much more natural. Alternatively, it would be interesting if there are other invariant polynomials besides the cumulants.



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