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Random fields of bounded variation and computation of their variation intensity

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  11 January 2017

Bruno Galerne
Bruno Galerne*
Affiliation:
Université Paris Descartes
*
* Postal address: Laboratoire MAP5 (UMR CNRS 8145), Université Paris Descartes, Sorbonne Paris Cité, 45 rue des Saints-Pères, 75006 Paris, France. Email address: [email protected]
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Abstract

The main purpose of this paper is to define and characterize random fields of bounded variation, that is, random fields with sample paths in the space of functions of bounded variation, and to study their mean total variation. Simple formulas are obtained for the mean total directional variation of random fields, based on known formulas for the directional variation of deterministic functions. It is also shown that the mean variation of random fields with stationary increments is proportional to the Lebesgue measure, and an expression of the constant of proportionality, called the variation intensity, is established. This expression shows, in particular, that the variation intensity depends only on the family of two-dimensional distributions of the stationary increment random field. When restricting to random sets, the obtained results give generalizations of well-known formulas from stochastic geometry and mathematical morphology. The interest of these general results is illustrated by computing the variation intensities of several classical stationary random field and random set models, namely Gaussian random fields and excursion sets, Poisson shot noises, Boolean models, dead leaves models, and random tessellations.

Keywords

Functions of bounded variationdirectional variationvariation intensityspecific perimeterstationary increment random fieldgerm–grain model

MSC classification

Primary: 60G60: Random fields
Secondary: 60G17: Sample path properties 60G51: Processes with independent increments; L'evy processes 60D05: Geometric probability and stochastic geometry
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 947 - 971
Copyright
Copyright © Applied Probability Trust 2017 

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