This is related to one of my previous questions [here](https://mathoverflow.net/questions/398449/k-textth-maxima-of-n-i-i-d-chi-square-random-variables).


Let $(Z\_1, Z\_2, \ldots, Z\_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I\_{n\times n} + \mu \boldsymbol{1}\_n\boldsymbol{1}\_n^\top $. Here $\boldsymbol{1}\_n$ denotes the vector of 1's of length $n$. Let us define $X\_i = Z\_i^2$ and I am trying to investigate the asymptotic of $k^{th}$ order statistic $X\_{(k:n)}$ where $k/n \to 1$ as $n \to \infty$.


As a special case let $k=n$. When $\mu =0$, it is known that $T\_n:=X\_{(n:n)}/\log n \overset{p}{\to} 2$.


What can we say about $T\_n$ when $0\leq \mu<1$? I have some conjectures about $T\_n$. If I am not wrong then one can use Slepian's lemma to conclude that $P(X\_{(n:n)}>t)\leq P(Y\_{(n:n)}>t)$ for $t\in \mathbb{R}$, where $\{Y\_i\}\_{i=1}^n$ are i.i.d $\chi^2\_1$. Though I am not completely sure of this fact. Here is the idea I am thinking about. Let $(W\_1, \cdots,W\_n)\sim N(0, I\_{n\times n})$. By Slepians' lemma we know $P(Z\_{(n:n)}> t)\leq P(W\_{(n:n)}>t)$. Now I am trying to take square of $Z\_{(n:n)}$ and $W\_{(n:n)}$. But squaring is not monotone and I am stuck. But I believe in asymptotic regime both of them are positive and hence my claim should work.


If this is true then I believe that $P(T\_n>2)\to 0$, i.e. heuristically speaking $T\_n \leq 2$ in limiting sense.


Any help will be appreciated. Also is any result for general $\Omega$ known? Thanks.



Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables
*(To long for a comment.)*


Let $b\_n = \sqrt{\frac{\mu}{n} + \frac{1-\mu}{n^2}} - \frac{\sqrt{1-\mu}}{n}$. Then


$$ \bigl( \sqrt{1-\mu} \, I\_{n\times n} + b\_n \mathbf{1}\_n \mathbf{1}\_n^{\top} \bigr)^2 = (1-\mu) I\_{n\times n} + \mu \mathbf{1}\_n \mathbf{1}\_n^{\top}. $$


In light of this, we can realize $(Z\_i)\_{1\leq i \leq n}$, using $(W\_i)\_{1\leq i\leq n} \sim \mathcal{N}(0,I\_{n\times n})$, by


$$ Z\_i = \sqrt{1-\mu} \, W\_i + b\_n S, $$


where $S=\sum\_{i=1}^{n} W\_i$. Then the law of iterated logarithm tells that $b\_n S = \mathcal{O}(\sqrt{\log\log n})$ almost surely, whereas the near-extreme values of $W\_i$'s will be of order $\sqrt{\log n}$.


For me, this seem to suggest that the near-maxima behavior of $(Z\_i^2)\_{1\leq i\leq n}$ will be very similar to that of the rescaled i.i.d. variables $((1-\mu)W\_i^2)\_{1\leq i\leq n}$.


