A Two-Step Markov Point Process

Existence and uniqueness are established for a translation-invariant Gibbs measure corresponding to a spatial point process that has, in addition to inhibition and clustering, the new feature of penalizing isolated points. It is shown that this point process has the so-called two-step Markov property, and a theorem is proved that characterizes the more general m-step Markov density functions. The asymptotic normality of certain statistics of the point process is proved when the size of the observation window tends to IR.