  The Hawkes process is a simple point process, whose intensity function
depends on the entire past history and is self-exciting and has the clustering
property. The Hawkes process is in general non-Markovian. The linear Hawkes
process has immigration-birth representation. Based on that, Fierro et al.
recently introduced a generalized linear Hawkes model with different exciting
functions. In this paper, we study the convergence to equilibrium, large
deviation principle, and moderate deviation principle for this generalized
model. This model also has connections to the multivariate linear Hawkes
process. Some applications to finance are also discussed.
