# zbMATH — the first resource for mathematics

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
Full Text: