Limit Theorems for an Extended Inverse Hawkes Process with General Exciting Functions

An inverse Hawkes process is a process having constant intensity and stochastic jump size, depending on the past number of jumps, while a Hawkes process has the intensity which is stochastic. An extended inverse Hawkes process is a process obtained by combining a Hawkes process and an inverse Hawkes process. The focus of this paper is to investigate the asymptotic behaviour of an extended inverse Hawkes process with general structure of the exciting functions. In particular, the results obtained are the generalized versions of the Law of Large Numbers and of the Central Limit Theorem.