The modelling of random bi-phasic or porous media (bones, stones, …) has been, and still is, subject to investigation by mathematicians, physicists or physicians. Here, we consider a thresholded random process X as a model, and look for information through the intervals when X is in a given phase, named chords. We focus on the study of the chord-length tail distribution function. In the literature, different types of tail’s behavior have been observed, among which exponential or power like decay. In this work, we look for the link between those two possible types of decay for the chord-length tail distribution function and the covariance function of X. First we perform with no a priori on the chord-length tail’s behavior, a statistical analysis on simulated data, using the Mean Excess Plot method as a graphical method to discriminate between light or heavy tails, and estimating the shape parameter of the associated GPD of the tail. Then we provide theoretical results proving on one hand the exponential decay of the chord-length tail distribution function when considering exponentially decreasing covariance function of a stationary Gaussian process , on the other hand, a decay faster than any negative power function when considering a r-mixing process X. Joint work with Y. Demichel, A. Estrade and G. Samorodnitsky
KRATZ, M. (2009). On the decay of Chord-lengths. Dans: Stochastic Processes and their Applications.