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How fast can the chord length distribution decay?

Part of:Stochastic processesSpecial processesGeometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Yann Demichel,
Anne Estrade,
Marie Kratz and
Gennady Samorodnitsky
Yann Demichel*
Affiliation:
Université Paris Ouest Nanterre La Défense
Anne Estrade*
Affiliation:
Université Paris Descartes
Marie Kratz*
Affiliation:
ESSEC Business School Paris
Gennady Samorodnitsky*
Affiliation:
Cornell University
*
∗ Postal address: MODAL'X, EA 3454, Université Paris Ouest Nanterre La Défense, 200 Avenue de la République, 92001 Nanterre Cedex, France. Email address: yann.demichel@u-paris10.fr
∗∗ Postal address: MAP5, UMR CNRS 8145, Université Paris Descartes, 45 rue des Saints-Pères, 75270 Paris 06, France. Email address: anne.estrade@parisdescartes.fr
∗∗∗ Postal address: ESSEC Business School Paris, Avenue Bernard Hirsch BP 50105, 95021 Cergy Pontoise Cedex, France. Email address: kratz@essec.fr
∗∗∗∗ Postal address: School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA. Email address: gennady@orie.cornell.edu
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Abstract

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The modeling of random bi-phasic, or porous, media has been, and still is, under active investigation by mathematicians, physicists, and physicians. In this paper we consider a thresholded random process X as a source of the two phases. The intervals when X is in a given phase, named chords, are the subject of interest. We focus on the study of the tails of the chord length distribution functions. In the literature concerned with real data, different types of tail behavior have been reported, among them exponential-like or power-like decay. We look for the link between the dependence structure of the underlying thresholded process X and the rate of decay of the chord length distribution. When the process X is a stationary Gaussian process, we relate the latter to the rate at which the covariance function of X decays at large lags. We show that exponential, or nearly exponential, decay of the tail of the distribution of the chord lengths is very common, perhaps surprisingly so.

Keywords

Chord lengthcrossingGaussian fieldbi-phasic mediumtail of distribution

MSC classification

Primary:60K05: Renewal theory
Secondary:60D05: Geometric probability and stochastic geometry60G10: Stationary processes60G55: Point processes60G70: Extreme value theory; extremal processes

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 2 , June 2011, pp. 504 - 523
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

During the elaboration of this work, Yann Demichel was a member of MAP5, Université Paris Descartes.

Marie Kratz is also a member of MAP5, Université Paris Descartes.

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