TITLE: Gaps between descending order statistics
QUESTION [7 upvotes]: Let $\{X_{1},X_{2},\cdots,X_{n}\}$ be a random sample of size $n$. Denote $(X_{(1)},X_{(2)},\cdots,X_{(n)})$ to be its descending order statistics. Define gap $g_{i}(n)$ to be $g_{i}(n)=X_{(i)}-X_{(i-1)},1\leq i\leq n$. My question is what is the limiting distribution of $g_{i}(n)$ as $n\to\infty$ or after some scaling if necessary. Are there any results on this problem?

REPLY [2 votes]: If the $X_i$ are independent and uniformly distributed on $[0,1]$, the (finite) point process $\sum_1^n \delta_{nX_i}$ converges in law to a Poisson point process on $\mathbb R^+$ with unit intensity, so the scaled gaps $ng_i(n)$ are exponentially distributed (density $e^{-x}$) in the limit.
More generally, if $F(y)=Pr\{X_i>y\}$ , the distribution of $ng_i(n)$ should converge to $\exp[-x/\ell_i]\ dx/\ell_i$ where $\ell_i=F'(F^{-1}(i/n))$, at least if $F$ is smooth enough (and $F'(F^{-1}(i/n))>0$). There are probably published results of this kind (Kolmogorov?...)