Sharp and Exact Tail Estimates for Multitype Poissonian Galton Watson Processes and Inhomogeneous Cluster Processes

TL;DR

This paper establishes sharp exponential tail estimates for multitype Poissonian Galton-Watson and Hawkes processes, providing tools for analyzing their probabilistic behavior especially in high-dimensional settings.

Contribution

It introduces new sharp exponential tail estimates for multitype Poissonian processes and applies these to improve understanding of multitype Hawkes processes and inhomogeneous clusters.

Findings

Derived exponential moments for multitype Galton-Watson processes.

Provided sharp tail estimates depending on offspring expectations and interaction decay.

Enhanced probabilistic analysis tools for high-dimensional Hawkes processes.

Abstract

We prove exponential moments for linear combinations of the number of individuals of each type of a whole multitype Poissonian Galton Watson process. We give sharp estimates for such quantities, which depend on the expectation of the Poissonian offspring distribution and the linear combination. We apply this result to derive type specific exponential moments for multitype Hawkes processes, improving existing results when the dimension is large. We also prove exponential estimates for tails of inhomogeneous Poissonian clusters. These estimates depend on the decay of the interaction functions defining the clusters and are starting points to derive probabilistic properties on branching processes such as Hawkes processes.