Summary
Let P be the distribution of a stationary point process on the real line and let P 0 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.
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Nieuwenhuis, G. Equivalence of functional limit theorems for stationary point processes and their Palm distributions. Probab. Th. Rel. Fields 81, 593–608 (1989). https://doi.org/10.1007/BF00367306
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DOI: https://doi.org/10.1007/BF00367306
Keywords
- Stochastic Process
- General Condition
- Probability Measure
- Probability Theory
- Stationary Point