# Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables

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.

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

• Thank you @Sangchui Lee. It looks like it does the job. Commented Aug 2, 2021 at 19:34
• @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. Commented Aug 2, 2021 at 20:00
• @Sangchui Lee: Sounds reasonable. thanks. Commented Aug 2, 2021 at 21:01