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Poisson thinning by monotone factors. (English) Zbl 1110.60050
Summary: Let \(X\) and \(Y\) be Poisson point processes on \(\mathbb{R}\) with rates \(\lambda_1, \lambda_2\) respectively. We show that if \(\lambda_1> \lambda_2\), then there exists a deterministic map \(\varphi\) with \(\varphi(X)\overset {d}=Y\) such that the joint distribution of \((X, \varphi(X))\) is translation-invariant and which is monotone in the sense that for all intervals \(I\), \(\varphi(X)(I)\leq X(I)\), almost surely.

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
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