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Stationary countable dense random sets

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Wilfrid S. Kendall
Wilfrid S. Kendall*
Affiliation:
University of Warwick
*
∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: [email protected]
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Abstract

We study the probability theory of countable dense random subsets of (uncountably infinite) Polish spaces. It is shown that if such a set is stationary with respect to a transitive (locally compact) group of symmetries then any event which concerns the random set itself (rather than accidental details of its construction) must have probability zero or one. Indeed the result requires only quasi-stationarity (null-events stay null under the group action). In passing, it is noted that the property of being countable does not correspond to a measurable subset of the space of subsets of an uncountably infinite Polish space.

Keywords

Hausdorff dimensionhitting setpoint processquasi-stationaritystationaritystochastic geometryzero-one law

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000 , pp. 86 - 100
Copyright
Copyright © Applied Probability Trust 2000 

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