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How Fast Can the Chord-Length Distribution Decay?

Author
Demichel, Yann; Estrade, Anne; Kratz, Marie; Samorodnitsky, Gennady
Abstract
The modelling of random bi-phasic, or porous, media has been, and still is,
under active investigation by mathematicians, physicists or 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, different types of the 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.
Sponsorship
French grant ``mipomodim''ANR-05BLAN-0017 ARO grant W911NF-07-1-0078
Date Issued
2009-09-21Subject
chord lengths; crossings; Gaussian fields; bi-phasic medium; tail of distribution
Type
technical report