We present sufficient conditions for sums of dependent point processes to converge in distribution to a Poisson process. This extends the classical result of Grigelionis [Theory Probab. Appl. 8 (1963) 172–182] for sums of uniformly null point processes that have Poisson limits. Included is an application in which a particle-survivor point process converges to a Poisson process. This result sheds light on the “surprising” Poisson limit of the species competition process of Durrett and Limic [Stochastic Process. Appl. 102 (2002) 301–309].
Publié le : 2007-02-14
Classification:
Poisson process,
sums of point process,
particle system,
Laplace functional,
convergence in distribution,
60G55
@article{1171377184,
author = {Jones, Matthew O. and Serfozo, Richard F.},
title = {Poisson limits of sums of point processes and a particle-survivor model},
journal = {Ann. Appl. Probab.},
volume = {17},
number = {1},
year = {2007},
pages = { 265-283},
language = {en},
url = {http://dml.mathdoc.fr/item/1171377184}
}
Jones, Matthew O.; Serfozo, Richard F. Poisson limits of sums of point processes and a particle-survivor model. Ann. Appl. Probab., Tome 17 (2007) no. 1, pp. 265-283. http://gdmltest.u-ga.fr/item/1171377184/