Remark on Van Lieshout and Baddeley's J-Function for Point Processes

摘要

The empty space function of a stationary point process in R(sup d) is thefunction which assigns to each r,r > 0, the probability that there is no point within distance r of O. It is natural to ask whether J = 1 implies that the point process is Poisson. The authors restrict to the one-dimensional case and show that a classical construction by Szasz provides an immediate counterexample. In this example, the interpoint distances are still exponentially distributed. This raised the question whether it is possible to have J = 1 but non-exponentially distributed interpoint distances. The authors construct a point process with J = 1 but where the interpoint distances are bounded.