Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-19T01:53:42.740Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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–1/2L 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
Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 1 , March 1990 , pp. 124 - 133
Copyright
Copyright © Applied Probability Trust 1990 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aldous, D., Flannery, B. and Palacios, J. L. (1988) Two applications of urn processes. Prob. Eng. Inf. Sci. 2, 293–307.CrossRefGoogle Scholar
Athreya, K. B., and Ney, P. E. (1972) Branching Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Bailey, N. (1964) The Elements of Stochastic Processes. Wiley, New York.Google Scholar
Billingsley, P. (1986) Probability and Measure, 2nd edn. Wiley, New York.Google Scholar
Chartrand, G. and Lesniak, L. (1986) Graphs and Diagraphs. Wadsworth, Monterey, CA.Google Scholar
Durrett, R. and Iglehart, D. L. (1977) Functionals of Brownian meander and Brownian excursion. Ann. Prob. 5, 130–135.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
Gupta, V. K. and Waymire, E. (1988) The spatial statistics of random networks and a problem in river basin hydrology. In Spatial Statistics and Imaging, ed. Possolo, A., IMS Lecture Notes – Monograph Series. To appear.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Prentice-Hall, Englewood Cliffs, NJ.CrossRefGoogle Scholar
Kendall, D. G. (1948) On the generalized birth and death processes. Ann. Math. Statist. 19, 1–15.Google Scholar
Kennedy, D. P. (1975) The Galton-Watson process conditioned on the total progeny. J. Appl. Prob. 12, 800–806.Google Scholar
Kolchin, V. F. (1978) Moment of degeneration of a branching process and height of a random tree Math. Notes 6, 954–961.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.Google Scholar
Mesa, O. J. and Gupta, V. K. (1987) On the main channel length-area relationship for channel networks. Water Resources Res. 23, 2119–2122.Google Scholar
Moon, J. W. (1980) On the expected diameter of random channel networks. Water Resources Res. 16, 1119–1120.Google Scholar
Nguyen, B. (1988) On the shape of the clusters of percolation of coalescing random walks. Preprint.Google Scholar
Pittel, B. (1986) Paths in a random digital tree: limiting distributions. Adv. Appl. Prob. 18, 139–155.Google Scholar
Troutman, B. M. and Karlinger, M. R. (1984) On the expected width function for topologically random channel networks. J. Appl. Prob. 22, 836–849.Google Scholar
Waymire, E. (1989) On the main channel length-magnitude formula for random networks: A solution to Moon's conjecture. Water Resources Res. 25, 1049–1050.Google Scholar