Let \u03a0 and \u0393 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 \u03a0 and \u0393 such that \u0393 is a deterministic function of \u03a0, and all points of \u0393 are points of \u03a0. 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 \u03a0 than in \u0393.<\/p>\n","protected":false},"excerpt":{"rendered":"
Let \u03a0 and \u0393 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 \u03a0 and \u0393 such that \u0393 is a deterministic function of \u03a0, and all points of \u0393 are points of \u03a0. 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