Suppose $X_1,X_2,...,X_n\sim f(x)$ where $f(x)$ is a pdf and even function, i.e. $f(x)=f(-x)$.
Now given that $$a_n(X_{(n:n)}-b_n)\xrightarrow{dist.} G_1$$ Then obviously from symmetry we can conclude that $$a_n(-X_{(n:1)}-b_n)\xrightarrow{dist.} G_2$$
where $G_1, G_2$ are standard Gumbel distribution.
Now I have to show,
$$a_n(X_{(n:n)}-b_n,-X_{(n:1)}-b_n)\xrightarrow{dist.} (G_1,G_2)$$
Where $G_1$ and $G_2$ are independent. But I do not know how proceed. Any kind of help appreciable.
There is some general result on it due to Gumbel Here. If anyone explain this proof simply that also appreciable.
Thanks in advance