I would like to investigate if the following quantity has already a well-known meaning in statistic:

$$G_{\eta}(X) =\frac{1}{\eta} \log\Biggl[1 + \lim_{N\to +\infty}\sum_{n=2}^N\frac{\eta^n}{n!}\mathbb{E}\bigl[(X -\mathbb{E}[X])^n\bigr]\Biggr] $$

where $X$ is a stochastic variable. I think than in general, for $\eta > 0$ higher $G(X)$ corresponds to higher entropy of $X$. But I'm not sure whether this last statement is true, and neither how to show it. the idea of entropy comes from

$$\frac{1}{\eta}\log \mathbb{E}[e^{\eta X}] = \mathbb{E}[X] + \frac{1}{\eta}\log \mathbb{E}[e^{\eta (X - \mathbb{E}[X])}] = \mathbb{E}[X] + G_{\eta}(X)$$

considering that $\frac{1}{\eta}\log \mathbb{E}[e^{\eta X}]$ is strictly connected to the dual of the entropy maximization problem.

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    $\begingroup$ You seem to have used $n$ as an index and as a limit variable (also the sum upper range is suspicious). Also, where does this come from? $\endgroup$ – Alex R. Mar 27 '18 at 17:16
  • $\begingroup$ This expression is very nearly the cumulant generating function. $\endgroup$ – whuber Mar 27 '18 at 17:24
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    $\begingroup$ Thanks Alex R. It was a mistake. Thanks Whuber for pointing out the cumulant generating functions. Seems to be what I was looking for. $\endgroup$ – Sam Mar 29 '18 at 9:38

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