  We study in this work the convergence of a sequence of asymptotically
critical self-exciting systems. In these systems, we use multivariate marked
Hawkes point processes to describe the event arrivals. Under mild assumptions
we prove the weak convergence of their rescaled density processes to a
multi-type continuous-state branching process with immigration (CBI-process).
In addition, we also provide two scaling limits for their shot noise processes
with limits being functionals of the multi-type CBI-process. Finally, we
provide a Hawkes representation for Crump-Mode-Jagers branching processes and
apply our limit results to study their asymptotic behavior. Specially, we
observe an interesting phenomenon, known as state space collapse in queueing
theory, in the population structure described by two measure-valued processes:
age-distribution process and residual-life distribution process. Indeed, under
a convergence condition on the life-length distributions, both the two rescaled
measure-valued processes converge weakly to a common measure-valued diffusion
process, which can be characterized as a lifting map of a multi-type
CBI-process associated with the limit excess life-length distribution.
