Let $X_1,X_2,...X_n$ be iid with the same chi-square distribution with one degree of freedom. Find $a_n$ and $b_n$ such that $a_n (\max_{1 \leq i\leq n}X_{i} - b_n)$ converges in distribution to a non-degenerate random variable.
Comments: I proved an inequality $\sqrt{\frac{2}{\pi}} \frac{t}{1+t^2}\exp({-\frac{t^2}{2}}) \leq P(|X| \geq t) \leq \sqrt{\frac{2}{\pi}} \frac{1}{t}\exp({-\frac{t^2}{2}})$ in the first part of the question, which I believe could be useful here, but I don't know how to apply here. I've seen several such questions so I appreciate any ideas or solutions to this type of questions.