Suppose that I have a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom $X_1, X_2, \ldots, X_n$, and denote $X_{\max}=\max(X_1, X_2, \ldots, X_n)$. Let $k$ be increasing but not as fast as $n$, i.e. $k=\omega(1)$ and $n=\omega(k)$.
I am wondering how far away $X_{\max}$ is from the rest of the sequence as $k\rightarrow\infty$. In particular, what is known about the asymptotics of the expected value $E[\Delta]$ where $\Delta$ is the fraction of the instances of the random variable in the sequence that are at least $\sqrt{k/\log(n)}$ less than $X_{\max}$ (formally, $\Delta=|\{X_i:X_i\leq X_{\max}-\sqrt{k/\log(n)}\}|/n$)? Is $\lim_{n\rightarrow\infty}E[\Delta]=0$ or $\lim_{n\rightarrow\infty}E[\Delta]=1$ (I doubt it's between zero and one)?
