Poisson limits of sums of point processes and a particle-survivor model

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].