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Clustering of Markov chain exceedances. (English) Zbl 1284.60106

Let \(X=\left( X_{0},X_{1},\dots\right) \) be a positive recurrent Markov chain on \(\left[ 0,\infty \right)\). The authors derive the limit of the exceedance point process defined by \[ N_{n}\left( \left[ 0,s\right] \times \left( a,\infty \right] \right) =\#\left\{ j\leq sn:X_{j}>ab_{n}\right\} , \] where \(b_{n}\rightarrow \infty \) is a threshold sequence. The chain \(X\) displays a regenerative stucture, so the sample path of \(X\) splits into identically distributed cycles between visits to certain set. Recall, the tail chain of a Markov chain is an asymptotic process that models behaviour upon reaching an extreme state. In this paper, the tail chain is used to describe the extremal behaviour of the regenerative cycles using their extremal components. Under some general conditions on extremal behaviour, the point process \(N_{n}\) converges to a cluster Poisson process, where the heights of the points in each cluster are determined by an independent run of the tail chain.

MSC:

60G70 Extreme value theory; extremal stochastic processes
60J05 Discrete-time Markov processes on general state spaces

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