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Deterministic thinning of finite Poisson processes. (English) Zbl 1208.60047
Summary: Let \( \Pi\) and \( \Gamma\) be homogeneous Poisson point processes on a fixed set of finite volume. We prove a necessary and sufficient condition on the two intensities for the existence of a coupling of \( \Pi\) and \( \Gamma\) such that \( \Gamma\) is a deterministic function of \( \Pi\), and all points of \( \Gamma\) are points of \( \Pi\). The condition exhibits a surprising lack of monotonicity. However, in the limit of large intensities, the coupling exists if and only if the expected number of points is at least one greater in \( \Pi\) than in \( \Gamma\).

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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