Multivariate Poisson Distributions Associated with Boolean Models

  • We consider a d-dimensional Boolean model Z = (Z_1+X_1) cup (Z_2+X_2) cup ... generated by a Poisson point process X_i, i = 1,2,... with some intensity measure and a sequence Z_i, i = 1,2,... of independent copies of some random compact set Z_0. Given compact sets K_1,...,K_l, we show that the discrete random vector (N(K_1),...,N(K_l)), where N(K_j) equals the number of shifted sets Z_i+X_i hitting K_j, obeys a l-variate Poisson distribution with 2^l-1 parameters. We obtain explicit formulae for all these parameters which can be estimated consistently from an observation of the union set Z in some unboundedly expanding window W_n (as n --> infty) provided that the Boolean model is stationary. Some of these results can be extended to unions of Poisson k-cylinders for k =1,...,d-1 and more general set-valued functionals of independently marked Poisson processes.

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Author:Christian Bräu, Lothar HeinrichGND
Frontdoor URL
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2015-10)
Publishing Institution:Universität Augsburg
Release Date:2015/08/03
Tag:random closed sets; independently marked Poisson process; generating functional; multivariate probability generating function; higher-order covariances; empirical volume fraction
GND-Keyword:Poisson-Prozess; Stochastische Geometrie; Multivariate Wahrscheinlichkeitsverteilung; Kovarianz <Stochastik>; Zufällige Menge; Geometrische Wahrscheinlichkeit
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Stochastik und ihre Anwendungen
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):License LogoDeutsches Urheberrecht mit Print on Demand