Asymptotic distributions for the intervals estimators of the extremal index and the cluster-size probabilities

Under appropriate long range dependence conditions, the point process of exceedances of a stationary sequence weakly converges to a homogeneous compound Poisson point process. This limiting point process can be characterized by the extremal index and the cluster-size probabilities. In this paper we address the problem of estimating these quantities and we consider the intervals estimators introduced in Ferro and Segers 2003. Inference for clusters of extreme values. J. Roy. Statist. Soc. Ser. B 545–556] and in Ferro 2004. Statistical methods for clusters of extreme values. Ph.D. Thesis, Lancaster University]. We establish asymptotic weak convergence to Gaussian random variables and we give their asymptotic variance.