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Extremal behaviour of models with multivariate random recurrence representation. (English) Zbl 1118.60060
The authors consider a $$q$$-dimensional stochastic recurrence equation $$Y_{n}=A_{n}Y_{n-1}+{\zeta}_{n}$$, $$n\in {\mathbb N}$$, for some iid sequence $$\{(A_{n},{\zeta}_{n})\}_{n\in{\mathbb N}}$$ of random $$q\times q$$-matrices $$A_{n}$$ and $$q$$-dimensional vectors $${\zeta}_{n}$$. The results of the paper concern the extremal behavior of the process $$y_{n}=z_{*}^{\prime} Y_{n}$$, $$n\in{\mathbb N}$$, for $$z_{*}\in{\mathbb R}^{q}$$ with $$| z_{*}| =1$$. They can be considered as an extension of results by L. de Haan, S. I. Resnick, H. Rootzén and C. G. de Vries [Stochastic Processes Appl. 32, No. 2, 213–224 (1989; Zbl 0679.60029)] and B. Basrak, R. A. Davis and T. Mikosch [ibid. 99, No. 1, 95–115 (2002; Zbl 1060.60033)] to the case where $$A_{n}$$ can be general $$q\times q$$-matrices. The paper starts with results on the existence of a stationary solution of the process $$\{Y_{n}\}_{n\in{\mathbb N}}$$. The main results present a description of the limit distribution of properly normalized running maxima $$M_{n}=\max\{y_{1},\dots,y_{n}\}$$, and give an explicit representation of the compound Poisson limit of the point processes of exceedances of $$\{y_{n}\}_{n\in{\mathbb N}}$$ over high thresholds in terms of its Poisson intensity and its jump distribution which represents the cluster behavior of such models on high levels. The obtained results are complemented by examples showing their consequences for a random coefficient autoregressive process.

##### MSC:
 60J05 Discrete-time Markov processes on general state spaces 60H25 Random operators and equations (aspects of stochastic analysis) 60G70 Extreme value theory; extremal stochastic processes 90B05 Inventory, storage, reservoirs 91B84 Economic time series analysis
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