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This is related to one of my previous questions here.

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.

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(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 $((1-\mu)W_i^2)_{1\leq i\leq n}$.

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  • $\begingroup$ Thank you @Sangchui Lee. It looks like it does the job. $\endgroup$
    – De vinci
    Commented Aug 2, 2021 at 19:34
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    $\begingroup$ @Devinci, Glad it helped! Also I just realized that we actually do not need LIL here, because we are essentially looking at the limit in distribution of $X_{(k;n)}/\log n$. So we may take advantage of the fact that $b_n S$ converges to a normal distribution by CLT. $\endgroup$ Commented Aug 2, 2021 at 20:00
  • $\begingroup$ @Sangchui Lee: Sounds reasonable. thanks. $\endgroup$
    – De vinci
    Commented Aug 2, 2021 at 21:01

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