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How fast can the chord length distribution decay?

Part of: Special processes Stochastic processes Geometric 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 geometry 60G10: Stationary processes 60G55: Point processes 60G70: Extreme value theory; extremal processes
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.

References

Baccelli, F. and Brémaud, P. (2003). Elements of Queueing Theory. Springer, Berlin.CrossRefGoogle Scholar
Bradley, R. C. (1986). Basic properties of strong mixing properties. In Dependence in Probability and Statistics (Oberwolfach, 1985; Progr. Prob. Statist. 11), Birkhäuser, Boston, MA, pp. 165–192.CrossRefGoogle Scholar
Bron, F. and Jeulin, D. (2004). Modelling a food microstructure by random sets. Image Anal. Stereology 23, 33–44.Google Scholar
Chaouche, A. and Bacro, J.-N. (2004). A statistical test procedure for the shape parameter of a generalized Pareto distribution. Comput. Statist. Data Anal. 45, 787–803.CrossRefGoogle Scholar
Cornfield, I. P., Fomin, S. V. and Sinai, Y. G. (1982). Ergodic Theory. Springer, New York.Google Scholar
Cramér, H. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. Sample Function Properties and Their Applications. John Wiley, New York.Google Scholar
Dietrich, C. R. and Newsam, G. N. (1997). Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput. 18, 1088–1107.Google Scholar
Doukhan, P. (1994). Mixing (Lecture Notes Statist. 85). Springer, New York.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (2003). Modelling Extremal Events. Springer, Berlin.Google Scholar
Estrade, A., Iribarren, I. and Kratz, M. (2008). Chord-distribution functions and Rice formulae. Application to random media. Available at http://hal.archives-ouvertes.fr/hal-00161806/fr/.Google Scholar
Gneiting, T. et al. (2006). Fast and exact simulation of large Gaussian lattice systems in R2: exploring the limits. J. Comput. Graph. Statist. 15, 483–501.Google Scholar
Hosking, J. R. M. and Wallis, J. R. (1987). Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29, 339–349.Google Scholar
Nott, D. J. and Wilson, R. J. (2000). Multi-phase image modelling with excursion sets. Signal Processing 80, 125–139.CrossRefGoogle Scholar
Pickands, J. (1975). Statistical inference using extreme order statistics. Ann. Statist. 3, 119–131.Google Scholar
Roberts, A. P. and Teubner, M. (1995). Transport properties of heterogeneous materials derived from Gaussian random fields: bounds and simulation. Phys. Rev. E 51, 4141–4154.Google Scholar
Roberts, A. P. and Torquato, S. (1999). Chord-distribution functions of three-dimensional random media: approximate first-passage times of Gaussian processes. Phys. Rev. E 59, 4953–4963.CrossRefGoogle ScholarPubMed
Schlather, M. (2001). Simulation and analysis of random fields. R News 1, 18–20.Google Scholar
Schlather, M. (2009). Reference manual for the Random Fields packages in R. Available at http://cran.r-project.org/web/packages/RandomFields/.Google Scholar
Schüth, F., Sing, K. S. W. and Weitkamp, J. (eds) (2002). Handbook of Porous Solids. John Wiley, New York.Google Scholar
Slepian, D. (1962). The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41, 463–501,Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications. John Wiley, Chichester.Google Scholar
Torquato, S. (2002). Random Heterogeneous Materials. Springer, New York.CrossRefGoogle Scholar