Exponential Tail Estimates for Multitype Poisson Branching Processes and Application to Hawkes Processes

Abstract

We establish exponential moment bounds for linear functionals of the total progeny of multitype Poisson Bienaymé-Galton-Watson trees. Our estimates are explicitly characterized in terms of the offspring mean matrix and the coefficients of the linear functional. As an application, we derive type-specific exponential moment bounds for multitype Hawkes processes, yielding improved results in high-dimensional settings. We also obtain exponential tail estimates for inhomogeneous Poisson clusters, with bounds that reflect the decay properties of the interaction kernels defining the clusters. These results provide useful probabilistic tools for the analysis of branching structures arising in Hawkes processes and related models.