Spatial point processes and the projection method

The projection method obtains non-trivial point processes from higher dimensional Poisson point processes by constructing a random subset of the higher dimensional space and projecting the points of the Poisson process lying in that set onto the lower dimensional region. This paper presents a review of this method related to spatial point processes as well as some examples of its applications. The results presented here are known for sometime but were not published before. Also, we present a backward construction of general spatial pure-birth processes and spatial birth and death processes based on the projection method that leads to a perfect simulation scheme for some Gibbs distributions in compact regions. Mathematics Subject Classification (2000). Primary 60G55, 60H99; Secondary 60J25, 60G40, 60G44.