Tail random variable and extremum Let $(X_n)_n$ be a sequence of real random variables. Is it true that $\limsup_n\frac{1}{n}\max_{1 \leq k\leq n}X_k$ is tail random variable? $\liminf_n \frac{1}{n}\min_{1 \leq k \leq n}X_k$ ? $\limsup_n\frac{1}{n}\min_{1\leq k \leq n}X_k$?
For the first two, I think the answer is correct, since for $r \in \mathbb{N}^*,$
$$\limsup_n\frac{1}{n}\max_{1 \leq k\leq n}X_k=\limsup_n\frac{1}{n+r}\max_{1\leq k \leq n+r}X_k=\limsup_n\max(\frac{1}{n+r}\max_{1\leq k \leq r}X_k,\frac{1}{n+r}\max_{r+1 \leq k \leq n+r}X_k)=\max(0,\limsup_n\frac{1}{n+r}\max_{r+1 \leq k\leq n+r}X_k)$$ which is $\sigma(\bigcup_{k \geq r}\sigma(X_k))$ measurable. 
(I used that $\limsup_n\max(u_n,v_n)=\max(\limsup_nu_n,\limsup_nv_n)$).
The same thing for $\liminf_n \frac{1}{n}\min_{1 \leq k \leq n}X_k.$
I'm stuck on the third one. Do you have any ideas?

In general: $$\left|\min\left(a,b\right)-\min\left(a',b\right)\right|\leq\left|a-a'\right|$$
So if: $$Z_{n}:=\frac{1}{n}\min_{2\leq k\leq n}X_{k}$$ and: $$Y_{n}:=\frac{1}{n}\min_{1\leq k\leq n}X_{k}=\min\left(\frac{1}{n}X_{1},Z_{n}\right)$$
then: $$\left|Y_{n}-\min\left(0,Z_{n}\right)\right|\leq\frac{1}{n}\left|X_{1}\right|$$
From this we conclude that: $$\limsup Y_{n}=\limsup\min\left(0,Z_{n}\right)$$
showing that $\limsup Y_{n}$ is measurable wrt $\sigma\left(X_{2},X_{3},\dots\right)$.
This can be expanded to find that $\limsup Y_{n}$ is measurable wrt
$\sigma\left(X_{r},X_{r+1},\dots\right)$ for every $r$.
