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November, 1984 Strong Limit Theorems for Maximal Spacings from a General Univariate Distribution
Paul Deheuvels
Ann. Probab. 12(4): 1181-1193 (November, 1984). DOI: 10.1214/aop/1176993147


Let $X_1, X_2, \cdots$ be an i.i.d. sequence of random variables with a continuous density $f$. We consider in this paper the strong limiting behavior as $n \rightarrow \infty$ of the $k$th largest spacing $M^{(n)}_k$ induced by $X_1, \cdots, X_n$ in the sample range. In the case where $f$ is bounded away from zero inside a bounded interval and vanishes outside, we characterize the limiting behaviour of $M^{(n)}_k$ in terms of the local behavior of $f$ in the neighborhood of the point where it reaches its minimum. In the case where the support of $f$ is an unbounded interval, we prove that for any $k \geq 1, M^{(n)}_k \rightarrow 0$ a.s. as $n \rightarrow \infty$ if and only if the distribution of $X_1$ has strongly stable extremes.


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Paul Deheuvels. "Strong Limit Theorems for Maximal Spacings from a General Univariate Distribution." Ann. Probab. 12 (4) 1181 - 1193, November, 1984.


Published: November, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0558.62018
MathSciNet: MR757775
Digital Object Identifier: 10.1214/aop/1176993147

Primary: 60F15

Keywords: Almost sure convergence , Empirical processes , Laws of the iterated logarithm , order statistics , Quantile processes , spacings , strong laws

Rights: Copyright © 1984 Institute of Mathematical Statistics

Vol.12 • No. 4 • November, 1984
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