How Fast Can the Chord-Length Distribution Decay?

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.