摘要：We study marked Hawkes processes with an intensity process which has a non-stationary baseline intensity, and a general self-exciting function of event ages at each time and marks. The marks are assumed to be conditionally independent given the even times, while the distribution of each mark depends on the event time, that is, time-varying. We first observe an immigration-birth (branching) representation of such a nonstationary marked Hawkes process, and then derive equivalent representation of the process using the associated conditional inhomogeneous Poisson processes with stochastic intensities. We consider such a Hawkes process in the high intensity regime, where the baseline intensity is scaled up, while the self-exciting function and distributions of the marks are unscaled, and there is no time-scaling in the scaled Hawkes process. We prove functional law of large numbers and functional central limit theorems (FCLT) for the scaled Hawkes processes in this asymptotic regime. The limits in the FCLT are characterized by continuous Gaussian processes with covariance structures expressed with convolution functionals resulting from the branching representation.