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Tree-dependent extreme values: the exponential case

Published online by Cambridge University Press:  14 July 2016

Vijay K. Gupta,
Oscar J. Mesa and
E. Waymire
Vijay K. Gupta*
Affiliation:
University of Colorado
Oscar J. Mesa*
Affiliation:
Universidad Nacional de Colombia
E. Waymire*
Affiliation:
Oregon State University
*
∗ Postal address: Center for the Study of Earth from Space/CERES and Department of Geological Sciences, University of Colorado, Boulder, CO 80309, USA.
∗∗ Postal address: Recursos Hidraulicos, Universidad Nacional de Colombia, Seccional de Medellin, Colombia.
∗∗∗ Present address: Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA.
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Abstract

The length of the main channel in a river network is viewed as an extreme value statistic L on a randomly weighted binary rooted tree having M sources. Questions of concern for hydrologic applications are formulated as the construction of an extreme value theory for a dependence which poses an interesting contrast to the classical independent theory. Equivalently, the distribution of the extinction time for a binary branching process given a large number of progeny is sought. Our main result is that in the case of exponentially weighted trees, the conditional distribution of n–1/2 L given M = n is asymptotically distributed as the maximum of a Brownian excursion. When taken with an earlier result of Kolchin (1978), this makes the maximum of the Brownian excursion a tree-dependent extreme value distribution whose domain of attraction includes both the exponentially distributed and almost surely constant weights. Moment computations are given for the Brownian excursion which are of independent interest.

Keywords

HYDROLOGYRIVER NETWORKBROWNIAN EXCURSION

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 1 , March 1990, pp. 124 - 133
Copyright
Copyright © Applied Probability Trust 1990 

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