Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T03:53:55.677Z Has data issue: false hasContentIssue false
  • English
  • Français

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: w.s.kendall@warwick.ac.uk
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aldous, D. and Barlow, M. T. (1981). On countable dense random sets. Lecture Notes in Math. 850, 311–327.Google Scholar
[2] Blackwell, D. and Dubins, L. E. (1975). On existence and non-existence of proper, regular, conditional distributions. Ann. Prob. 3, 741–752.Google Scholar
[3] Dellacherie, C. and Meyer, P. A. (1978). Probabilities and Potential A. North-Holland, Amsterdam.Google Scholar
[4] Federer, H. (1969). Geometric Measure Theory. Springer, Berlin.Google Scholar
[5] Halmos, P. R. (1974). Measure Theory. Springer, Berlin. [First edition 1950.].Google Scholar
[6] Himmelberg, C. J. (1975). Measurable relations. Fundamenta mathematicae 87, 53–72.Google Scholar
[7] Himmelberg, C. J., Parthasarathy, T. and Van Vleck, F. S. (1981). On measurable relations. Fundamenta mathematicae 111, 161–167.Google Scholar
[8] Hoffmann-Jörgensen, J., (1994). Probability with a View towards Statistics, Vols I, II. Chapman and Hall, New York.Google Scholar
[9] Ioffe, A. D. (1978). Survey of measurable selection theorems: Russian literature supplement. SIAM J. Control Optimization 16, 728–732.Google Scholar
[10] Kallenberg, O. (1977). A counterexample to {R. Davidson's} conjecture on line processes. Math. Proc. Camb. Phil. Soc. 82, 301–307.Google Scholar
[11] Kendall, D. G. (1974). Separability and measurability for stochastic processes: a survey. In Stochastic Analysis, eds Kendall, D. G. and Harding, E. F.. John Wiley, New York, pp. 415–443.Google Scholar
[12] Loomis, L. H. (1953). An Introduction to Harmonic Analysis. Van Nostrand, Toronto.Google Scholar
[13] Lusin, N. (1972). Leçons sur les ensembles analytiques et leurs applications. Chelsea, New York, 2nd (corrected) edn. [First edition Paris 1930.].Google Scholar
[14] Mackey, G. W. (1976). The Theory of Unitary Group Representations. University of Chicago Press, Chiacago, IL.Google Scholar
[15] Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, Chichester, UK.Google Scholar
[16] Mattila, P. (1985). On the {Hausdorff} dimension and capacities of intersections. Mathematika 32, 213–217.CrossRefGoogle Scholar
[17] Robbins, H. E. (1944). On the measure of a random set I. Ann. Math. Statist. 15, 70–74.Google Scholar
[18] Robbins, H. E. (1945). On the measure of a random set II. Ann. Math. Statist. 16, 342–347.Google Scholar
[19] Rudin, W. (1966). Real and Complex Analysis. McGraw-Hill, New York.Google Scholar
[20] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications, 2nd edn. John Wiley, Chichester, UK.Google Scholar
[21] Wagner, D. H. (1977). Survey of measurable selection theorems. SIAM J. Control Optimization 15, 859–903.Google Scholar
[22] Wagner, D. H. (1980). Survey of measurable selection theorems: an update. In Measure Theory Oberwolfach 1979, ed. Kölzow, D. ({Lecture Notes in Math.} 794). Springer, Berlin, pp. 176–219.Google Scholar
[23] Warner, F. W. (1983). Foundations of Differentiable Manifolds and Lie Groups. Springer, New York.CrossRefGoogle Scholar
[24] Zuyev, S. (1999). Stopping sets: {gamma}-type results and hitting properties. Adv. Appl. Prob. 31, 355–366.Google Scholar