Functional limiting behaviour of non-stationary marked Hawkes processes under multi-scaling high-intensity regime

RESUMO

This paper studies the non-stationary marked Hawkes processes in which the base intensity function is time-dependent and the kernel function is governed by an external random factor called mark. The marks are assumed to be determined by random events occurrence over time resulting in a non-identical distribution. These facts introduce two kinds of difficulties: one from the absence of the i.i.d. (independent and identical distributed) and the other is from the non-stationarity. In this framework, an asymptotic regime, often referred to as the high intensity regime, is considered, under which the intensity increases with time. This can be obtained by multiplying the time parameter by n alpha, for some alpha > 0. Under the high intensity regime, the functional law of large numbers and the functional central limit theorem are established. The rescaled (centered and normalized) marked Hawkes process is proved to converge in distribution to a Gaussian process, which is the accumulation of Gaussian noise and a diffusion process. In addition, a shot-noise process is studied, and the functional Limit Theorems under appropriate hypotheses are established.