Asymptotic theory for the moment-based parameter estimator

Establish the consistency, asymptotic distribution, dynamical properties, and dependence properties of the moment-based parameter estimator for the spatio-temporal coarse-grained Hawkes process.

Background

The spatio-temporal coarse-grained Hawkes process is defined through first- and second-order conditional moment properties, and its parameters are estimated by minimizing a moment-based loss derived from an autoregressive representation. Because the model does not specify a complete conditional probability law, the estimator is not based on an exact likelihood.

The paper derives approximation results for the stationary mean and autocovariance and demonstrates the practical performance of the estimator, including in an aggregated ETAS application. However, it does not prove that the estimator is consistent, characterize its asymptotic distribution, or establish the estimator's dynamical and dependence properties. These results are needed to provide a formal statistical foundation for inference based on aggregated spatio-temporal event counts.