Modeling clusters of extreme values and applications

We study clusters of rare events, namely, exceedances of the sequence over a sufficiently high threshold. Such clusters are caused by dependence in time series. Such clusters of rare events and the asymptotic distributions of the cluster and inter-cluster sizes have been widely studied due to numerous applications, see Ancona-Navarrete and Tawn (2000), Beirlant et al. (2004), Ferro and Segers (2003), Markovich (2014), Robert (2009), (2013), Robinson et al. (2000) among others. We define the clusters as conglomerates containing consecutive threshold exceedances of the process separated by return intervals with consecutive non-exceedances. We establish geometric-like asymptotical distributions of the cluster and inter-cluster sizes and corresponding expectations. Rates of convergence of the distribution tail of the duration of clusters and return intervals between clusters to stable distribution tail are derived. The talk is based on recent author results. Distributions of the cluster and inter-cluster sizes of some example processes are presented. Applications of the results in social and telecommunication systems are discussed.