Some non asymptotic tail estimates for Hawkes processes

Abstract

We use the Poisson cluster process structure of a Hawkes process to derive non asymptotic estimates of the tail of the extinction time, of the coupling time or of the number of points per interval. This allows us to define a family of independent Hawkes processes; each of them approximating the initial process on a particular interval. Then we can easily derive exponential inequalities for Hawkes processes which can precise the ergodic theorem.