Which PDF of X leads to a Gumbel distribution of the finite-size average of X? Consider the statistic "average of $N$ idd random variables $X_i$",
$$S_N = \frac{1}{N} \sum_{i=1}^N X_i
$$
Consider also that, by a numerical experiment, it is observed that the distribution of $S_N$, $P(S_N)$, is a Gumbel distribution (e.g. p-value does not reject the null hypothesis that it is).
Which $P(X_i) = P(X)$ can explain these observations?
More quantitatively, what can lead to a Gumbel distribution of the (finite-size) mean of a random variable? If no such process exists for iid, what correlation can explain these observations?
(I am aware of the occurrence of the Gumbel distribution on block maxima, but this seems to be unrelated to this problem.)
EDIT:
In my actual problem, I currently do not have access to the time series of $X_i$, so I can't tell you whether they are correlated, but most likely they are correlated. Moreover, $X_i \in [0, 1]$. The iid assumption was a first approach to the problem, to see if there was a simple distribution of $X_i$ that could explain the Gumbel distribution of the mean.
 A: The Gumbel distribution has as pdf
$$f(x)=\exp\{-x-e^{-x}\}$$
which corresponds to the transform 
$$X=-\log(Y)\qquad Y\sim\mathcal{E}(1)$$
where $\mathcal{E}(\lambda)$ is the exponential distribution. The moment generating function is 
$$\Phi(t)=\mathbb{E}[\exp\{tX\}]=\Gamma(1-t)$$
by a simple change of variable into $Y$. As detailed in this book (p.443), albeit with a slightly different definition of the Gumbel, the Gamma function satisfies the property that
$$\Gamma(1-t)=e^{-\gamma t}\prod_{i=1}^\infty\left(1-\frac{t}{i}\right)^{-1}e^{-t/i}$$
where $\gamma$ is Euler's constant. The central term in the product is the moment generating function of an exponential variate $E_i$ with scale $1/i$. Therefore a Gumbel variable can be written as
$$X=\lim_{m\to\infty} \sum_{i=1}^m\{E_i-i^{-1}\}-\gamma=\lim_{m\to\infty} \sum_{i=1}^m E_i-\log(m)$$
It then follows from the infinite decomposability of the exponential distribution that each $E_i$ can be written as $E_i=\sum_{j=1}^n V_{ij}/n$, where the $V_{ij}$'s are Gamma $\mathcal{G}(1/n,n/i)$, hence that
$$X=\lim_{m\to\infty} \sum_{i=1}^m \left[ \sum_{j=1}^n \{V_{ij}/n-\log(m)/n\}\right]=\frac{1}{n}\sum_{j=1}^n \left[\lim_{m\to\infty} \sum_{i=1}^m \{V_{ij} - \log(m)\}\right]$$
is indeed infinitely divisible. (Thanks to Gérard Letac for his help.)
A: To complement Xi'an's answer, here is a situation where one encounters the Gumbel distribution as a sum of a random variable (not a mean).
Let independent random variables $X_j=-\log(V_j)$, where  $V_j\sim\mbox{beta}(j\sigma,1-\sigma)$, $j\in\mathbb{N}^*$, $\sigma\in(0,1)$. Then the sum of $X_j$s, after proper centering, behaves very similarly to a Gumbel distribution, see for instance the plot here: Central limit theorem for independent random variables, with a Gumbel limit.
