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Independence of thinned processes characterizes the Poisson process: an elementary proof and a statistical application. (English) Zbl 1119.62089
Summary: Let $$N, N_{1}$$ and $$N_{2}$$ be point processes such that $$N_{1}$$ is obtained from $$N$$ by homogeneous independent thinning and $$N_{2}=N - N _{1}$$. We give a new elementary proof that $$N_{1}$$ and $$N_{2}$$ are independent if and only if $$N$$ is a Poisson point process. We also present an application of this result to test if a homogeneous point process is a Poisson point process.

##### MSC:
 62M30 Inference from spatial processes 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 62M07 Non-Markovian processes: hypothesis testing 62P10 Applications of statistics to biology and medical sciences; meta analysis
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