The Annals of Probability

On the Influence of the Extremes of an I.I.D. Sequence on the Maximal Spacings

Paul Deheuvels

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Abstract

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.

Article information

Source
Ann. Probab. Volume 14, Number 1 (1986), 194-208.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992622

Digital Object Identifier
doi:10.1214/aop/1176992622

Mathematical Reviews number (MathSciNet)
MR815965

Zentralblatt MATH identifier
0594.60029

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems

Keywords
Order statistics spacings extreme values weak convergence limiting distribution

Citation

Deheuvels, Paul. On the Influence of the Extremes of an I.I.D. Sequence on the Maximal Spacings. Ann. Probab. 14 (1986), no. 1, 194--208. doi:10.1214/aop/1176992622. https://projecteuclid.org/euclid.aop/1176992622


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