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January, 1986 On the Influence of the Extremes of an I.I.D. Sequence on the Maximal Spacings
Paul Deheuvels
Ann. Probab. 14(1): 194-208 (January, 1986). DOI: 10.1214/aop/1176992622


Let $X_1, X_2,\cdots$ be an i.i.d. sequence of random variables with a continuous density $f$, positive on $(A, B)$, and null otherwise. Under the assumption that $Y_n = \min\{X_1,\cdots, X_n\}$ and $Z_n = \max\{X_1,\cdots, X_n\}$ belong to the domain of attraction of extreme value distributions and that $f(x) \rightarrow 0$ as $x \rightarrow A$ or $x \rightarrow B$, we show that the weak limiting behavior of $Y_n$ and $Z_n$ characterizes completely the weak limiting behavior of the maximal spacing generated by $X_1,\cdots, X_n$ and obtain the corresponding limiting distributions. We study as examples the cases of the normal, Cauchy, and gamma distributions.


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Paul Deheuvels. "On the Influence of the Extremes of an I.I.D. Sequence on the Maximal Spacings." Ann. Probab. 14 (1) 194 - 208, January, 1986.


Published: January, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0594.60029
MathSciNet: MR815965
Digital Object Identifier: 10.1214/aop/1176992622

Primary: 60F15

Keywords: Extreme values , Limiting distribution , order statistics , spacings , weak convergence

Rights: Copyright © 1986 Institute of Mathematical Statistics


Vol.14 • No. 1 • January, 1986
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