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A Remark on the Van Lieshout and Baddeley J-Function for Point Processes

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

T. Bedford  and
J. Van Den Berg
T. Bedford*
Affiliation:
Delft University of Technology
J. Van Den Berg*
Affiliation:
CWI
*
∗ Postal address: Delft University of Technology, Faculty of Technical Mathematics and Informatics, Mekelweg 4, 2628 CD Delft, The Netherlands.
∗∗ Postal address: CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands.
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Abstract

The empty space function of a stationary point process in ℝd is the function that assigns to each r, r > 0, the probability that there is no point within distance r of O. In a recent paper Van Lieshout and Baddeley study the so-called J-function, which is defined as the ratio of the empty space function of a stationary point process and that of its corresponding reduced Palm process. They advocate the use of the J-function as a characterization of the type of spatial interaction.

Therefore it is natural to ask whether J ≡ 1 implies that the point process is Poisson. We restrict our analysis to the one-dimensional case and show that a classical construction by Szász provides an immediate counterexample. In this example the interpoint distances are still exponentially distributed. This raises the question whether it is possible to have J ≡ 1 but non-exponentially distributed interpoint distances. We construct a point process with J ≡ 1 but where the interpoint distances are bounded.

Keywords

POINT PROCESSJ-FUNCTIONEMPTY SPACE FUNCTIONPOISSON PROCESS

MSC classification

Primary: 60G55: Point processes 60J05: Discrete-time Markov processes on general state spaces 60G10: Stationary processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 29 , Issue 1 , March 1997 , pp. 19 - 25
Copyright
Copyright © Applied Probability Trust 1997 

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References

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