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Poisson Thinning by Monotone Factors

   
@article{ECP1134,
	author = {Karen Ball},
	title = {Poisson Thinning by Monotone Factors},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {10},
	year = {2005},
	keywords = {},
	abstract = {Let $X$ and $Y$ be Poisson point processes on the real numbers with rates $l_1$ and $l_2$ respectively.  We show that if $l_1 > l_2$, then there exists a deterministic map $f$ such that $f(X)$ and $Y$ have the same distribution, the joint distribution of $(X, f(X))$ is translation-invariant, and which is monotone in the sense that for all intervals $I$, $f(X)(I) \leq X(I)$, almost surely.},
	pages = {no. 7, 60-69},
	issn = {1083-589X},
	doi = {10.1214/ECP.v10-1134},    
        url = {http://ecp.ejpecp.org/article/view/1134}}