I was reading the paper "on the minimum of several random variables". In example 10 item (ii) it states:
Let $1\leq k\leq n$. Let $g_i,i\leq n$, be independent $N(0,1)$ Gaussian random variables. Then for $k\leq n/2$: $$c\sqrt{\ln\frac{2n}{n+1-k}}\leq\mathbb{E}(k\text{-}\min_{i\leq n}|g_i|)\leq C\sqrt{\ln\frac{2n}{n+1-k}}$$ where $c,C$ are absolute constants.
The author claimed that this is a well-known inequality so he didn't provide the proof. I am interested whether this inequality has a name and how this was proved. Especially I am interested in the proof techniques, since recently I was working on a project related to this topic, I would like to see whether or not similar result can be generalized to other distributions( e.g. sub-gaussian).
Does anyone know any reference related to this? Thanks a lot.
A clarification: The notation $(k\text{-}\min_{i\leq n}|g_i|)$ is used to mean "the $k$-th smallest value of $|g_i|$ over the set of $n$ variates." (mf)
