A Remark on the Van Lieshout and Baddeley J-Function for Point Processes
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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.
[1] Isaac Meilijson,et al. A characterization of marginal distributions of (possibly dependent) lifetime variables which right censor each other , 1997 .
[2] T. Mattfeldt. Stochastic Geometry and Its Applications , 1996 .
[3] A. Baddeley,et al. A non-parametric measure of spatial interaction in point patterns , 1996, Advances in Applied Probability.
[4] Daryl J. Daley,et al. An Introduction to the Theory of Point Processes , 2013 .