Transforming Spatial Point Processes into Poisson Processes

Transforming spatial point processes into Poisson processes Frederic Schoenberg*,^ Department of Statistics, University of California at Los Angeles, Los Angeles, CA 90095, USA Abstract In 1986, Merzbach and Nualart demonstrated a method of transforming a two-parameter point process into a planar Poisson process of unit rate, using random stopping sets. Merzbach and Nualart's theorem applies only to a special class of point processes, since it requires two restrictive conditions: the (F4) condition of conditional independence and the convexity of the 1-compensator. The (F4) condition was removed in 1990 by Nair, but the convexity condition remained. Here both the (F4) condition and the convexity condition are removed by making use of predictable sets rather than stopping sets. As with Nair's theorem, the result extends to point processes in higher dimensions. Keywords: Compensator; intensity; point process; Poisson process; predictable set; random space change; spatial process; stopping time Introduction. Suppose A^ is a point process. Is it possible to rescale the domain in such a way that A is transformed into a Poisson process with rate 1? * Research supported by Michel and Line Loeve fellowship and by the University of California Campus- Laboratory Collaboration program Advanced Earthquake Hazard Research . tEmail: frederic@stat.ucla.edu

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