Study of discrete-time Hawkes process and its compensator

The discrete-time Hawkes process (DTHP) is a sub-class of $g$-functions that serves as a discrete-time version of the continuous-time Hawkes process (CTHP). Like the CTHP, the DTHP also has the self-exciting property and its intensity depends on the entire history. In this paper, we study the asymptotic behaviour of the DTHP and its compensator. We further analyse the moment generating function (MGF) of the DTHP and obtain some bounds and convergence results on the scaled logarithmic MGF of the DTHP.