Sufficient conditions for long-range count dependence of stationary point processes on the real line

Daley and Vesilo (1997) introduced long-range count dependence (LRcD) for s tationary point processes on the real line as a natural augmentation of the classical long-range dependence of the corresponding interpoint sequence. They studied LRcD for some renewal processes and some output processes of q ueueing systems, continuing the previous research on such processes of Dale y (1968), (1975). Subsequently, Daley (1999) showed that a necessary and su fficient condition for a stationary renewal process to be LRcD is that unde r its Palm measure the generic lifetime distribution has infinite second mo ment. We show that point processes dominating, in a sense of stochastic ord ering, LRcD point processes are LRcD, and as a corollary we obtain that for arbitrary stationary point processes with finite intensity a sufficient co ndition for LRcD is that under Palm measure the interpoint distances are po sitively dependent (associated) with infinite second moment. We give many e xamples of LRcD point processes, among them exchangeable, cluster, moving a verage, Weld, semi-Markov processes and some examples of LRcD point process es with finite second Palm moment of interpoint distances. These examples s how that, in general, the condition of infiniteness of the second moment is not necessary for LRcD. It is an open question whether the infinite second Palm moment of interpoint distances suffices to make a stationary point pr ocess LRcD.