I was going through the book Information, Physics and Computation by Mezard and Montanari, where in Chapter 8, I found the following relation:
\begin{equation} \mathbb{E} \log Z = \lim_{n\rightarrow 0} \frac{1}{n} \log (\mathbb{E} Z^n)~, \end{equation}
where $Z$ is the partition function of the random energy model (REM) (to be precise, $Z = \sum_{j=1}^{2^N} \exp(-\beta E_j)$, where $E_1,\cdots,E_{2^N}$ are i.i.d. $N(0,N/2)$).
My question is that, does the above relation hold for any non-negative random variable $Z$, or is it a specific property of the REM partition function? I am confused, because I have not seen this relation anywhere before.
Many thanks.