Strong-mixing rates for Hawkes processes and application to Whittle estimation from count data

Felix Cheysson (1) , Gabriel Lang (1)

1 MIA Paris-Saclay - Mathématiques et Informatique Appliquées

In this paper, we study the time series generated by the event counts of the stationary Hawkes process. Using the cluster properties of the stationary Hawkes process, we prove an upper bound for its strong-mixing coefficient, and for its count series', provided that the reproduction kernel has a finite (1 + β)-th order moment (for a β > 0). When the exact locations of points are not observed, but only counts over fixed time intervals, we propose a spectral approach to the estimation of Hawkes processes, based on Whittle's likelihood. This approach provides consistent and asymptotically normal estimates provided common regularity conditions on the reproduction kernel. Simulated datasets illustrate the performances of the estimation, notably, of the Hawkes reproduction mean and kernel, even with relatively large time intervals.

Mots clés: Hawkes process; strong mixing; count data; Whittle method; spectral estimation

Domaines: Statistiques [math.ST]

Felix Cheysson, Gabriel Lang. Strong-mixing rates for Hawkes processes and application to Whittle estimation from count data. 2021. ⟨hal-03117924v1⟩