Approximate distributions of clusters of extremes. (English) Zbl 1095.62063

Let \(\{X_n,n \geq 1\}\) be a stationary sequence of random variables. Let \(\{u_n\}\) be a real sequence such that \(P(X_1>u_n)>0\) but \(P(X_1>u_n) \to 0\) as \(n \to \infty\). For a positive integer \(m\), define \(M_m=\max \{X_1,\ldots,X_m\}\). The author studies exceedances over the threshold \(\{u_n\}\) among the variables \(X_1,\ldots,X_{r_n}\), where \(\{r_n\}\) is positive integer sequence tending to infinity. He derives approximations of the distribution of a cluster of exceedances conditionally on \(\{M_{r_n}>u_n\}\) in terms of the distribution of \((X_1,\ldots,X_m)\) conditionally on \(\{X_1>u_n\}\) (the tail chain approximation), and in terms of the conditional distribution of \((X_1,\ldots,X_{2m})\) conditionally on \(X_{m+1}\) being the cluster peak, that is, \(X_{m+1}=M_{2m}>u_n\) (the cluster peak approximation). A brief discussion of possible statistical applications is also given.


62G32 Statistics of extreme values; tail inference
60G70 Extreme value theory; extremal stochastic processes
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