# Exponential distribution as limiting distribution

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.

-

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.