## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 43, Number 2 (2011), 504-523.

### How fast can the chord length distribution decay?

Yann Demichel, Anne Estrade, Marie Kratz, and Gennady Samorodnitsky

#### Abstract

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.

#### Article information

**Source**

Adv. in Appl. Probab., Volume 43, Number 2 (2011), 504-523.

**Dates**

First available in Project Euclid: 21 June 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1308662490

**Mathematical Reviews number (MathSciNet)**

MR2848388

**Zentralblatt MATH identifier**

1222.60068

**Subjects**

Primary: 60K05: Renewal theory

Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G10: Stationary processes 60G55: Point processes 60G70: Extreme value theory; extremal processes

**Keywords**

Chord length crossing Gaussian field bi-phasic medium tail of distribution

#### Citation

Demichel, Yann; Estrade, Anne; Kratz, Marie; Samorodnitsky, Gennady. How fast can the chord length distribution decay?. Adv. in Appl. Probab. 43 (2011), no. 2, 504--523. https://projecteuclid.org/euclid.aap/1308662490