Some non asymptotic tail estimates for Hawkes processes

Some non asymptotic tail estimates for Hawkes processes Patricia Reynaud-Bouret? and Emmanuel Roy† 10th January 2006 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. MSC Classification: 60G55. Keywords: Point processes, exponential inequalities, approximate simulation of a stationary Hawkes process. Introduction The Hawkes processes have been introduced by Hawkes (1971). Since then they are especially applied to earthquake occurrences (Vere-Jones 1970), but have recently found applications to DNA modeling (Gusto & Schbath 2005). In particular, an assumption which was not very realistic for earthquakes is very reasonable in this framework: the support of the reproduction measure is known and bounded. The primary work is motivated by getting non asymptotic concentration inequalities for the Hawkes process, using intensively the bounded support assumption. Those con- centration inequalities are fundamental to construct adaptive estimation procedure as the penalized model selection (Massart 2000, Reynaud-Bouret 2003). To do so, we study intensively in this paper the link between cluster length, extinction time and construction of an approximating family of independent processes.

• derive tail estimates


• hawkes process


• then


• galton-watson process


• consider independently


• sub-critical galton-watson process


• consider now independently


• process structure


• reproduction measure


• poisson cluster