The independence of ordered distances

Question

Let $X_1,\dots,X_n$ be i.i.d. random variables, for a fixed $X_k$ let $D_{(1:n)},\ldots,D_{(n:n)}$ be the order statistics of the random variables $d(X_1,X_k),\dots,d(X_n,X_k)$ (which are the distances between $X_k$ and the other random variables). My problem is that (for a fixed $k$) I cannot conclude anything about distribution of the ordered distances $D_{(1:n)},\ldots,D_{(n:n)}$ since the underlying $D_1,\ldots,D_n$ are not independent. Can they somehow asymtotically become independent in addition to being identically distributed as $n \to \infty$? Does it maybe help to only look at $P(D_i < a, D_j < b)$ for very small $a,b$? Is there anything one can do to have any notion of independence for $D_1,\ldots, D_n$?