# Convergence almost sure of the minimum of a succession of random variables

I have a problem with this exercise.

Let $(X_n)$ be an IID succession such that $X_n$ ~ $\mathcal{N}(0,1)$ and set $Y_n := n\min(|X_1|, \dots, |X_n|)$. Can anyone explain me intuitively why $Y_n$ does not converge almost surely? And what can be said about $\liminf_{n \to \infty} Y_n$?

I have already shown that $Y_n$ converges to $\text{Exp}(\sqrt{\frac{2}{\pi}})$ in distribution.

• How did you show that $Y_n$ converges in distribution? I found a more general statement of this problem here: lpsm.paris//pageperso/merle/M1.td2.12.pdf and am curious if the limit distribution is exponential regardless of the distribution of the $X_n$. – Math1000 Jun 6 '18 at 16:00