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The tail empirical process of regularly varying functions of geometrically ergodic Markov chains. (English) Zbl 1448.60114

Summary: We consider a stationary regularly varying time series which can be expressed as a function of a geometrically ergodic Markov chain. We obtain practical conditions for the weak convergence of the tail array sums and feasible estimators of cluster statistics. These conditions include the so-called geometric drift or Foster-Lyapunov condition and can be easily checked for most usual time series models with a Markovian structure. We illustrate these conditions on several models and statistical applications. A counterexample is given to show a different limiting behavior when the geometric drift condition is not fulfilled.

MSC:

60G70 Extreme value theory; extremal stochastic processes
62G32 Statistics of extreme values; tail inference
60F05 Central limit and other weak theorems
60G10 Stationary stochastic processes
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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