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

Diffusion Approximations for Marked Self-exciting Systems with Applications to General Branching Processes