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I wonder if there are well-known and studied cases involving Exponential distribution as limiting distribution. I also wonder if this would contradict Central Limit theorem.

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Since exponential distribution is a special case of Weibull distribution, and that arises as a limit distribution in extreme value theory (see Gnedenko-Fisher-Tippet theorem).

For example, consider a uniform (on a unit interval) sample of size $n$. Then $$ \lim_{n \to \infty} \mathbb{P}\left( \min(u_1,u_2,\ldots,u_n) \leqslant \frac{x}{n}\right) = \lim_{n \to \infty} 1- \left(1-\frac{x}{n}\right)^n = 1 - \exp(-x) $$ The latter is exactly the CDF of the exponential random variable with unit mean.

This does not contradict the CLT as the latter does not apply for this case.

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