Let $Z \sim \mathcal{N}(0, I)$. Let $Z_{(k)}$ be the $k$th order statistic of $Z$.

  1. Is it true that $\text{Var}(Z_{(k)}) \to 0$ as $n\to \infty$ for $1 \leq k \leq n$?
  2. Any estimate on the rate?

What about for some more general $Z \sim \mathcal{N}(\mu, \Sigma)$?

  • $\begingroup$ @Xi'an Almost. Checking the references in the answer, I cant see a way to bound the remainder terms in the expansion. $\endgroup$ – Student Mar 18 '20 at 23:04
  • $\begingroup$ Can anyone explain how the linked "duplicate" answers my question? The link provides an estimate on the variance based on a Taylor expansion, ignoring the higher order terms. It's clear that the truncated version is asymptotically zero, but the contribution of the higher order terms is completely ignored. $\endgroup$ – Student Mar 19 '20 at 22:54