let's say $\underline{X}_n=(X^1_n,\ldots,X^k_n)$ is a series of random vectors, each consisting of $k$ i.i.d. entries, which converges in distribution to $\underline{X}$ as $n$ grows to infinity. We could analyse these vectors as samples of length $k$.
Clearly, a la Skorokhod, the first order statistic $X^{[1]}_n\xrightarrow{d}X^{[1]}$ then also converges in distribution, since taking the maximum of a vector is a continuous operation from $\mathbb{R}^k\to\mathbb{R}$. Compare $\max(v_1,v_2)=(v_1+v_2+|v_1-v_2|)/2$. The same could be said about the $k$-th order statistic, aka the minimum. But what about all the order statistics in between, can we say anything about their convergence?
I'm aware of what makes these two especially easy: for the maximum to be $\leq x$ is the same event as having all entries be $\leq x$. Meanwhile for the order statistics in between you'd have to do some combinatorics. But still, I couldn't find anything on this and would be thankful for all help :)