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

Abstract

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.

A Ray-Knight Theorem for Spectrally Positive Stable Processes

“…• The local time process of a compound Poisson process with drift −1 stopped at hitting 0 is equal in distribution to a homogeneous, binary Crump-Mode-Jagers process (CMJ-process) starting from one ancestor; see [29]. • A homogeneous, binary CMJ-process can be reconstructed as the density process of a marked Hawkes point measure with arrivals and marks of events representing the birth times and life-lengths of offsprings; see [21,37].…”

“…A functional central limit theorem has been established for multivariate Hawkes processes in [2] and for Marked Hawkes point measures in [21] respectively. For the nearly unstable Hawkes process with light-tailed kernel, Jaisson and Rosenbaum [24] proved the weak convergence of the rescaled intensity to a Feller diffusion, which was generalized to multivariate marked Hawkes point processes and their shot noise processes in [37]. For the heavy-tailed case, they also proved that the rescaled point process converges weakly to the integral of a rough fractional diffusion; see [13,25].…”