Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-08T19:29:24.968Z Has data issue: false hasContentIssue false
  • English
  • Français

On weak stationarity and weak isotropy of processes of convex bodies and cylinders

Part of: General convexity Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Lars Michael Hoffmann
Lars Michael Hoffmann*
Affiliation:
Universität Karlsruhe (TH)
*
∗ Postal address: Institut für Algebra und Geometrie, Universität Karlsruhe (TH), 76128 Karlsruhe, Germany. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Generalized local mean normal measures μz, z ∈ Rd, are introduced for a nonstationary process X of convex particles. For processes with strictly convex particles it is then shown that X is weakly stationary and weakly isotropic if and only if μz is rotation invariant for all z ∈ Rd. The paper is concluded by extending this result to processes of cylinders, generalizing Theorem 1 of Schneider (2003).

Keywords

Particle processcylinder processPoisson processmean normal measureweak stationarityweak isotropy

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 39 , Issue 4 , December 2007 , pp. 864 - 882
Copyright
Copyright © Applied Probability Trust 2007 

References

Fallert, H. (1996). Quermassdichten für Punktprozesse konvexer Körper und Boolesche Modelle. Math. Nachr. 181, 165–184.CrossRefGoogle Scholar
Hoffmann, L. M. (2007). Intersection densities for nonstationary Poisson processes of hypersurfaces. Adv. Appl. Prob. 39, 307–317.CrossRefGoogle Scholar
Hug, D. and Last, G. (2000). On support measures in Minkowski spaces and contact distributions in stochastic geometry. Ann. Prob. 28, 796–850.CrossRefGoogle Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press.CrossRefGoogle Scholar
Schneider, R. (2003). Nonstationary Poisson hyperplanes and their induced tessellations. Adv. Appl. Prob. 35, 139–158.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (1992). Integralgeometrie. Teubner, Stuttgart.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.CrossRefGoogle Scholar
Weil, W. (1997). Mean bodies associated with random closed sets. Rend. Circ. Mat. Palermo (2) Suppl. 50, 387–412.Google Scholar
Weil, W. (1997). On the mean shape of particle processes. Adv. Appl. Prob. 29, 890–908.CrossRefGoogle Scholar