Erratum to: Modeling clusters of extreme values

Ordinal patterns in clusters of subsequent extremes of regularly varying time series

“…An advantage of the definition of uexceedance clusters is that it depends on one parameter, namely the threshold u, only. Such a cluster definition has already been employed in a series of papers by Markovich (2014Markovich ( , 2016Markovich ( , 2017 who analyzes the limit distribution of two cluster characteristics. First, she considers the number of inter-cluster times T 1 (u), i.e.…”

“…we have lim u→∞ P(T 2 (u) = 1) = 1. In Markovich (2014Markovich ( , 2016, under appropriate mixing conditions, the rate of convergence is determined as a function of the extremal index. More precisely, for all ε > 0, there exist a threshold u 0 = u 0 (ε) a number j 0 = j 0 (ε) such that, for all u > u 0 and j > j 0 , 1 − P(X 0 > u) θ θ 2 (1 − P(X 0 > u))…”

“…that is, for a sufficiently large threshold u, the tail of the distribution of T 2 (u) becomes approximately proportional to a geometric distribution with parameter 1 − P(X 0 > u) θ . Furthermore, Markovich (2014Markovich ( , 2016 provides results for the duration of clusters if the time between subsequent observations is random with regularly varying distribution.…”

Clustering and hitting times of threshold exceedances and applications

“…Due to dependence such exceedances tend to occur in clusters. 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), Markovich (2016a), Robert (2009), Robert (2013), Roberts et al (2006), Robinson et al (2000) among others. There are three approaches in the cluster size study, namely, the blocks method, the runs method and the inter-exceedance times method.…”

“…non-ergodic sequences with θ = 0 is given in Theorem 4 by Doukhan et al (2015). Using achievements regarding the limit geometric-like distribution of T 1 (x ρn ) derived in (Theorem 2, Markovich (2014), Markovich (2016a)), where the (1 − ρ n )th quantile x ρn of {X n } is taken as u n , we derive in Section 2 a limit distribution of the first hitting time and its expectation that specifies (6). The achievements are similarly extended to the second hitting time, Section 3.…”