Poisson point process: Difference between revisions

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===Poisson-type random measures===
 
The [[Poisson-type random measures]] (PT) are a family of three random counting measures which are closed under restriction to a subspace, i.e. closed under [[Point process operation#Thinning]]. These random measures are examples of the [[mixed binomial process]] and share the distributional self-similarity property of the [[Poisson random measure]]. They are the only members of the canonical non-negative [[power series]] family of distributions to possess this property and include the [[Poisson distribution]], [[negative binomial distribution]], and [[binomial distribution]]. The Poisson random measure is independent on disjoint subspaces, whereas the other PT random measures (negative binomial and binomial) have positive and negative covariances. The PT random measures are discussed<ref>Caleb Bastian, Gregory Rempala. Throwing stones and collecting bones: Looking for Poisson-like random measures, Mathematical Methods in the Applied Sciences, 2020. [[doi:10.1002/mma.6224]]</ref> and include the [[Poisson random measure]], negative binomial random measure, and binomial random measure.
 
===Poisson point processes on more general spaces===