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?



Gaps between descending order statistics
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?...)


