## 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.

### MSC:

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

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