The rescaled marked Hawkes point measure with immigration converges to a Gaussian white noise plus a Brownian-motion lifting, and the shot noise converges to a Brownian martingale.
Scaling Limits for Crump-Mode-Jagers Processes with Immigration via Stochastic Volterra Equations
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abstract
In this paper, we firstly give a reconstruction for Crump-Mode-Jagers processes with immigration as solutions to a class of stochastic Volterra integral equations, which offers us a new insight for the evolution dynamics of age-dependent population. Based on this new representation, we prove the weak convergence of rescaled Crump-Mode-Jagers processes with immigration to a class of continuous-state branching processes with immigration. Moreover, the limits reveal that the individual law mainly changes the branching mechanism and immigration mechanism proportionally. This covers the results obtained by Lambert et al. [35] for subcritical binary Crump-Mode-Jagers processes.
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Functional Limit Theorems for Marked Hawkes Point Measures
The rescaled marked Hawkes point measure with immigration converges to a Gaussian white noise plus a Brownian-motion lifting, and the shot noise converges to a Brownian martingale.