Limit Theorems for Inverse Process T_n of Hawkes Process

We consider linear Hawkes process N_t and its inverse process T_n. The limit theorems for N_t are well known and studied by many authors. In this paper, we study the limit theorems for T_n. In particular, we investigate the law of large numbers, the central limit theorem and the large deviation principle for T_n. The main tool of the proof is based on immigration-birth representation and the observations on the relation between N_t and T_n.