Let $X_1,...,X_n$ be i.i.d. random variables and $X_{(1)}<...<X_{(n)}$ be the order statistics. Assume $X_i\sim \text{exp}(1)$. Find a sequence of constants $a_n$ such that $X_{(n)}-a_n$ converges in distribution.
CDF of $X_{(n)}-a_n$ is $(1-e^{-(x+a_n)})^n$. What condition should we have over $a_n$ such that $(1-e^{-(x+a_n)})^n$ has a limit?
I think if $a_n=\ln(n)$ then $$(1-e^{-(x+a_n)})^n=(1-\frac{e^{-x}}{n})^n\rightarrow e^{-e^{-x}}$$
Correct. For your choice of $a_n$, $$\lim_{n \to \infty} X_{(n)} - a_n$$ has what is called an extreme value distribution, with CDF $e^{-e^{-x}}$ and the convergence is in distribution.
As an exercise, as a function of $n$, what is $\operatorname{E}[X_{(n)}]$? What happens if we choose $a_n = \operatorname{E}[X_{(n)}]$? What is the asymptotic variance in each case?