Optimality Properties of Empirical Estimators for Multivariate Point Processes

A multivariate point process is a random jump measure in time and space. Its distribution is determined by the compensator of the jump measure. By an empirical estimator we understand a linear functional of the jump measure. We give conditions for a nonparametric version of local asymptotic normality of the model as the observation time tends to infinity, assuming that the process either has no fixed jump times or is a discrete-time process. Then we show that an empirical estimator is efficient for the associated linear functional of the compensator of the jump measure. The latter functional is, in general, random. Under our conditions, its limit is deterministic. For homogeneous and positive recurrent processes, the limit is an expectation under the invariant distribution. Our result can be viewed as a first step in proving that the estimator is efficient for this expected value. We apply the result to Markov chains and Markov step processes.