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?
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2$\begingroup$ I think that the paper of Luc Devroye, Laws of iterated logarithm for order statistics of uniform spacings, Ann. Probab., 9(1981), 860-867 might help you. $\endgroup$– Liviu NicolaescuCommented Sep 28, 2015 at 14:42
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1$\begingroup$ Thank you so much, Liviu. It's a great honor for me to get your help. I think I listened your talk once at McMaster University in Hamilton. $\endgroup$– Youzhou ZhouCommented Sep 29, 2015 at 10:19
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2 Answers
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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?...)
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$\begingroup$ Thank you so much, I agree with you in the case where $X_{1},\cdots,X_{n}$ are $i.i.d.$ uniform random variables. $\endgroup$ Commented Sep 29, 2015 at 10:15
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You could look at books on extreme value theory like
1) Extreme Value Theory- An Introduction by De Haan and Ferreira (chapter 2)
2) http://link.springer.com/book/10.1007%2F978-3-0348-0009-9
Cheers
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$\begingroup$ Or perhaps it is not a complete answer. If you could point to specific results in those books which address the OP's question, instead of telling him/her to read an entire book or two, it really would be a proper answer. $\endgroup$ Commented Sep 28, 2015 at 8:21
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$\begingroup$ Sorry for not being specific. I think this paper by Weissman can help Op's question, " Estimation of Parameters and Large Quantiles Based on kth largest observations" by Ishay Weissman (look into theorem 3) $\endgroup$– RakshithCommented Sep 28, 2015 at 15:41