Equivalence of functional limit theorems for stationary point processes and their Palm distributions

Let P be the distribution of a stationary point process on the real line and let P0 be its Palm distribution. In this paper we consider two types of functional limit theorems, those in terms of the number of points of the point process in (0, t] and those in terms of the location of the nth point right of the origin. The former are most easily obtained under P and the latter under P 0. General conditions are presented that guarantee equivalence of either type of functional limit theorem under both probability measures, and under a third, P 1, which plays a role in the proofs and is obtained from P by shifting the origin to the first point of the process on the right.In a brief final section the obtained results for either type of functional limit theorem are extended to equivalences between the two types by applying well-known results about processes drifting to infinity and the corresponding inverse processes.