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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.

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    $\begingroup$ Since for large $N$ the Central Limit Theorem implies $S_N$ will have an approximately Normal distribution, and no Gumbel distribution is Normal, it is important to know the value of $N$. In particular, the answer when $N=1$ is a Gumbel distribution. Could you tell us what your $N$ is and what the estimated Gumbel parameters might be? $\endgroup$
    – whuber
    May 26, 2016 at 14:47
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    $\begingroup$ The Gumbel distribution is infinitely divisible, so there exists such a representation. $\endgroup$
    – Xi'an
    May 26, 2016 at 14:49
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    $\begingroup$ @Xi'an (It's ok to ignore the exponential in the CF because that's easily handled.) For $n\gt 1$ the root of the Gamma function is not a Gamma function, so you have just shown that the distribution of $X$ is not Gumbel. I don't recognize this CF, nor can I invert it. $\endgroup$
    – whuber
    May 26, 2016 at 14:57
  • $\begingroup$ @whuber: The central limit theorem does not apply when $X_i$ doesn't have its first two moments, which would narrow this down. $\endgroup$
    – Alex R.
    May 26, 2016 at 17:26
  • $\begingroup$ @J.C.Leitão: Are you absolutely positive your $X_i$ are iid? Could there be some correlation, or more precisely could the mean and variance of $X_i$ be changing with $i$? $\endgroup$
    – Alex R.
    May 26, 2016 at 17:27

2 Answers 2

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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.)

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  • $\begingroup$ I have made some corrections, but I still don't get the last step. I.e. How come that $\log(m)/n$ appears in the last equation? Replacing $E_i$ on the equation for X should not modify the term $\log(m)$. $\endgroup$ May 27, 2016 at 9:17
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    $\begingroup$ What about $\log(m)=\sum_{i=1}^n\log(m)/n$? $\endgroup$
    – Xi'an
    May 27, 2016 at 9:36
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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.

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