Convergence in distribution of $n \cdot \inf(X_1, ..., X_n)$

Question

Let $(X_n)_{n \in \mathbb N}$ be a sequence of i.i.d. random variables.

Let's suppose that $X_1$ follows a uniform distribution on [0, 1]. Does $n \cdot \inf(X_1, ..., X_n)$ converge in distribution? If yes, how can this be proved?

Additionnally, I'm given that $\forall a < x < b, \ \mathbb P(X_1 \geq x) < 1$.

Thanks for any help!

I assume you mean $0\leq a<x<b \leq 1.$ You could look here: https://math.stackexchange.com/a/1493441 — 311411, Jun 06 '21 at 14:53

score 0 · Accepted Answer · answered Jun 06 '21 at 15:55

Let $Y_{n}=n\cdot inf\left\{X_1,X_2,\ldots,X_n\right\}$ then

$$ 1-F_{Y_n}(y)=\prod_{i=1}^{n}\left(1-F_{X_{n}}(y/n)\right)\implies F_{Y_n}(y)=1-(1-y/n)^n $$

Taking the limit as $n\to\infty$

$$ F_Y(y)=1-e^{-y} $$ Si it converges in distribution to an exponential

Convergence in distribution of $n \cdot \inf(X_1, ..., X_n)$

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