Suppose $X_{1}, \ldots, X_{n}\overset{i.i.d.}{\sim} \mathcal{N}(0,1)$. Consider the order statistics $X_{(1)} \geq X_{(2)} \geq \ldots \geq X_{(n)}$. Then the Gaussian maxima $X_{(1)}$ has sub-gaussian type concentration: Tail bounds for maximum of sub-Gaussian random variables. The median $X_{(\lceil n/2 \rceil)}$ also has sub-gaussian type concentration: Concentration inequality for the median. But the dominating concentration result for general order statistics $X_{(k)}$, $k\neq 1$ or $\lceil n/2 \rceil$ only gives sub-exponential type of tail bound (Berinstein-like inequalities), c.f.:
Boucheron, Stéphane Vincent; Thomas, Maud, Concentration inequalities for order statistics, Electron. Commun. Probab. 17, Paper No. 51, 12 p. (2012). ZBL1349.60021.
I am wondering is the Bernstein-like inequaility the sharpest concentration for general order statistics of Gaussian? Can $X_{(k)}$ also have sub-gaussian concentration for general $k$?
