Let $\{X_i\}$ be a sequence of independent and identically distributed random variables, with distribution $N(\mu,\sigma^2)$. Define the random variables $$ Y_n=\frac{\max\{X_1,\dots,X_n\}-\min\{X_1,\dots,X_n\}}{n}. $$
Does $\{Y_n\}$ converge? If yes, to what? If not, is there a power of $n$ (at the denominator) for which it converges to a non-zero number almost certainly? By numerical experimentation, it seems that with $n$ it converges to $0$, but with $\log n$ it stays apart from $0$.
I was wondering whether one can recover $\sigma$ with this limit process.
It's equivalent (by simmetry) to finding the limit of $$\frac{2\cdot\max\{X_1,\dots,Xn\}}{n}$$ which is easier to compute.