Asymptotic properties of the maximum likelihood estimator for spatio-temporal point processes

Consider a spatio-temporal point process whose events occur at times in the interval CO, T1 and at corresponding locations in a region X. Such processes can be modeled through their conditional intensity function Lambda((s) under tilde,t;<(theta)under tilde>); 0 less than or equal to t less than or equal to T,(s) under tilde epsilon X. This article shows that the maximum likelihood estimator <(theta)under tilde>(T)<(theta)over cap> is consistent and asymptotically normally distributed as T --> infinity. These results extend those of Ogata (Ann. Inst. Statist. Math. 30A (1978), 243-261), who considered purely temporal pointproceses. The asymptotic properties of <(theta)under tilde>(T)<(theta)under tilde> are considered for a spatiotemporal self-exiciting point process. Methods for modeling spatio-temporal point patterns are illustrated on seismological data.