I am interested in the "asymptotic" covariance between the "top quartile" of order statistics. Formally speaking, given a sample of $n$ iid variables $X_1, \cdots X_n$ with finite second moment, consider the order statistics $X_{(1)}<\cdots< X_{(n)}$, and the following quantity where $p>0$:
$$\frac{1}{|\{i\neq j;\; i,j>pn\}|}\sum_{i\neq j;\; i,j>pn}^n Cov(X_{(i)},X_{(j)})$$
In words, this is the "average" covariance between all pairs of order statistics that are ranked at least $pn$. I am specifically interested to show that this quantity converges to 0 with a speed of $O(\frac{1}{n})$. The reason for this hypothesis is that this can be shown quite easily when $X_i$ are uniform random variables, as we know in this case that for $i,j = O(n)$: $$Cov(X_{(i)},X_{(j)})=\frac{j(n-k+1)}{(n+1)^2(n+2)}=O(\frac{1}{n})$$ And therefore we have, for any $p>0$: $$\frac{1}{|\{i\neq j;\; i,j>pn\}|}\sum_{i\neq j;\; i,j>pn}^n Cov(X_{(i)},X_{(j)}) = O(\frac{1}{n})$$ My question is how to extend this result in this case to where $X_i$ is a general random variable (with finite second moment). Any help would be greatly appreciated!!