# Deterministic Thinning of Finite Poisson Processes - Mathematics > Probability

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Abstract: Let Pi and Gamma be homogeneous Poisson point processes on a fixed set offinite volume. We prove a necessary and sufficient condition on the twointensities for the existence of a coupling of Pi and Gamma such that Gamma isa deterministic function of Pi, and all points of Gamma are points of Pi. Thecondition exhibits a surprising lack of monotonicity. However, in the limit oflarge intensities, the coupling exists if and only if the expected number ofpoints is at least one greater in Pi than in Gamma.

Author: ** Omer Angel, Alexander E. Holroyd, Terry Soo**

Source: https://arxiv.org/