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.


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